Short Answer:
Basic Evocation Wizards are now the equivalent of Tier 3 at lower levels.
Basic Champion Fighters are now the equivalent of Tier 3 at lower levels.

The Usefulness of Tiers will have to change for 5e, as tiers 1, 5, and 6 at this time don't exist, and likely won't exist as the overall power of 5e is much flatter. We can compare classes by looking at their average DPAD (damage per adventuring day).

For overall power however, it's going to really depend on level.

Thanks to the work by @Waxeagle we can see how power via damage shifts over level.

Basically at level 11, the Fighter is doing more damage than the evocation wizard who is focused solely on damage. However, by level 17, both the cleric and the Wizard have enough spell slots, to out perform the Fighter and Rogue. In addition, level 9 spells are by their nature "game changers". It should be noted that at level 20, with the third extra attack, the fighter shoots up in power as well, though not enough to out perform spells such as meteor storm, the nuclear weapon of D&D. One caveate to the current graph is that not all levels are shown, only the level when the "adventure tier" changes. For example, not shown on the graph is level 10, where the fighter still only has one extra attack, the wizard and cleric are doing about the same amount as the fighter.

So while it is still the case at the highest levels that magic is more powerful than the mundane, up until the epic tier, this problem does not really present itself.

This obviously doesn't take into account the non-damaging effects of spells, such as haste, or invisibility, or flying, etc. However, with the new concentration mechanic, these spells are situational and can not be easily piled on top of each other as they could in 3rd edition, to create a quadratic power curve.

Relevant data for the chart:
The above chart is based on the various classes an unlimited number of enemies with the following defenses, and no assumption is made about the HP of the monster, so damage is "unbounded" in the terms of the spreadsheet:
Dex Save 2
AC 15
Con Save 2

However, If I change the defences of the monsters to something less reasonable in the game, such as giving a Dex and Con save of 5 and an AC of 10, we see the fighter keeping up with the wizard and again surpassing it at level 20.

You can compare these 5e graphs, to this 3.5 graph and notice how the fighter is at the bottom of the graph and stays very linear, while the wizard has a slight a quadradic curve in power gains.

(Graph taken from here)

Old Long Answer about Tiers:

First, the assertion that a Wizard might be tier 1:

Tier 1: Capable of doing absolutely everything, often better than classes that specialize in that thing. Often capable of solving encounters with a single mechanical ability and little thought from the player. Has world changing powers at high levels. These guys, if played well, can break a campaign and can be very hard to challenge without extreme DM fiat, especially if Tier 3s and below are in the party.

It's pretty clear that currently, the Evocation Wizard can not do absolutely everything. Because of the concentration mechanic, the Wizard can really only do one cool thing at a time. You can't fly and be invisible at the same time without another magic item or wizard casting those spells on you, for example. Secondly, while some spells can duplicate other class abilities, such as "knock", those spells are only situationally better, and come at a high resource spot (limited spell slots) that other classes lack. So the Wizard is no longer a tier 1 class.

Instead, I would say that Tier 3 fits the Evocation Wizard.

Tier 3: Capable of doing one thing quite well, while still being useful when that one thing is inappropriate, or capable of doing all things, but not as well as classes that specialize in that area. Occasionally has a mechanical ability that can solve an encounter, but this is relatively rare and easy to deal with. Challenging such a character takes some thought from the DM, but isn't too difficult. Will outshine any Tier 5s in the party much of the time.

The bolded part is why I believe a Wizard is Tier 3. A wizard can, if picked the right spells, do anything another class can do (with some exceptions), but it can't do those things as well as the class that specializes in that thing. An invisible wizard is hidden, but can't sneak. A wizard with damage spells, can't kill, a single high hitpoint target as well as a fighter can. A Rogue will have better skill abilities when it specializes in those skills than a wizard can ever get.

Next the assertion that a fighter is tier 5.

Tier 5 is defined by:

Tier 5: Capable of doing only one thing, and not necessarily all that well, or so unfocused that they have trouble mastering anything, and in many types of encounters the character cannot contribute. In some cases, can do one thing very well, but that one thing is very often not needed. Has trouble shining in any encounter unless the rest of the party is weak in that situation and the encounter matches their strengths. DMs may have to work to avoid the player feeling that their character is worthless unless the entire party is Tier 4 and below. Characters in this tier will often feel like one trick ponies if they do well, or just feel like they have no tricks at all if they build the class poorly.

No class is good at only one thing. Because of the Background system, all classes and characters are able to focus on skills that are outside of their class. So while you can make a fighter that is only good at fighting, and can't do anything during exploration or social interactions, it's also more likely to build a fighter that is good at fighting and leading people. Or good at fighting and using certain tools. Because of bounded Accuracy, all characters are decent at whatever skill they are proficient in. The champion fighter is best built for two handed heavy weapons, but they can also be a good archer, or duelist. Fighters excel in combat and at higher levels, can move across the battle field, hitting multiple creatures in one turn in the same way that a wizard might target more than one creature with a spell. They are also able to survive situations that others would not without a cleric. So I don't think they can be called Tier 5.

Champion fighters are however tier 3, because they are

Capable of doing one thing quite well, while still being useful when that one thing is inappropriate.

Fighters do combat really well, they survive well and can kill well. However sometimes a situation will be called where there is no combat. Fighters can excel at jumping across chasms, or climbing/swimming out of tough spots more so than wizards and clerics. With Background again, they can also call upon their traits. In the starter set for example, the Fighter is a folk hero, who is able to rally others to their cause.

However, the tier system as a whole is a bad way to judge D&D 5e classes. The power level of the entire system has been made "more flat", and there will need to be better definitions of how to compare classes.

Fighter, Monk, Wizard, Sorcerer, Cleric, Shugenja (from Complete Divine), Rogue and Ninja (from Complete Adventurer)

Unfortunately, you have selected a massively disparate subset of the classes. Fighter, monk, ninja, and rogue are four of the weakest classes published for all of 3.5 (with Complete Adventurer’s ninja actually probably the worst here), while cleric, shugenja, sorcerer, and wizard are four of the strongest classes published (especially the cleric and wizard).

And ultimately, those spellcasters are so powerful that no one without spellcasting is going to be able to keep up if they’re played at anything like their potential.

If you are going to be selecting a very-limited subset of the classes, I suggest that you take the opportunity to choose better designed classes which have tighter overall balance. Your warriors are not powerful enough, your spellcasters are too powerful.

You can ignore the advice for everyone but the fighter and the monk – I still do strongly recommend the swordsage and warblade there – but even those improved options won’t be able to keep up with those spellcasters, at least not unless the people playing them are underplaying their power quite a lot. Therefore, I’d like to offer a more “complete” recommendation for every class on your list.

In all cases, I have carefully chosen the replacement class so that it can be a “drop-in” replacement – you don’t have to make up a new place in your setting for warblades, just use the warblade’s stats and rules and call it “fighter.”

Stronger warriors

The more “direct” answer to the question, there definitely are superior options for playing a “fighter” or “monk” than the fighter class or monk class. Note, however, that even these are still a far cry from the spellcasters you have selected (and, indeed, still are not quite up to the ability of even the weaker spellcasters I recommend below).

Instead of fighter, consider the warblade

The warblade from Tome of Battle: The Book of Nine Swords is an intelligent master-of-arms, a savvy tactician and a well-honed veteran of battle. His ability to master weapons is without parallel, and in battle he is relentless and iconic, rallying his allies around him.

Instead of monk or ninja, consider the swordsage

Also from Tome of Battle, the swordsage could be monk or ninja, depending on the particular choices that you make. A “monk-like” swordsage would probably focus more on Diamond Mind and Setting Sun maneuvers, while a “ninja-like” swordsage would likely focus on Shadow Hand and Tiger Claw – but either would be remiss to ignore those other disciplines.

Instead of rogue, consider the factotum

Dungeonscape’s factotum is basically “the class Indiana Jones would take if he were in D&D.” Factota do a little bit of everything, and do it well. They use their wits and cunning to make it up as they go along, and are the best skill-monkeys in the game, bar none. Awesome, super-fun class.

Weaker Spellcasters

I wrote an answer dedicated to this question here, but because that question was overly-broad, my answer was overly-long.

But the idea is still sound: the over-powered spellcasters you have chosen could be replaced quite well by other spellcasters that fulfill similar roles and have similar narrative backgrounds. Note that these spellcasters are still very powerful, however; using these is insufficient to make the fighter, monk, or ninja classes play on even footing. Even the superior classes I recommend above are at something of a disadvantage compared to most of these; the gap is just much narrower.

For this game, I strongly recomment the following replacements:

Instead of cleric, consider the divine bard

Bards are viable warriors, while not being excellent at it. The cleric would make a far better warrior than any of your warrior classes. The bard has some solid healing spells, especially when divine, and their music makes them fantastic leaders-of-men.

Alternatively, the incarnate might be excellent

The incarnate from Magic of Incarnum is an excellent replacement, flavor-wise, for a cleric, with a strong religious theme (an evil-aligned incarnate is supposed to literally be evil incarnate). Their powers lends themselves a bit more to the skill-monkey/support role than the warrior/healer role, but they’re overall an excellent class and judiciously awarding wands of cleric spells can make up a huge part of the difference.

If you specifically wanted the warrior-priest, crusader is for you

I generally recommend replacing cleric with classes that are less devastating at armed combat, because I tend to feel that the cleric’s prowess there makes it difficult to justify the existence of, e.g., paladins. But since you have not selected paladins in the first place, you might be looking for a more martial bent from your divine class. If so, the crusader from Tome of Battle: Book of Nine Swords is perfect – he heals his allies as he smites his enemies, and as a rallying point and commander of battle, he knows few peers.

Also, crusader is one of my favorite classes in the game, and my #1 recommendation to new players.

Instead of shugenja, consider the ... actually, shugenja’s probably OK

Shugenja is the weakest of the spellcasting classes that you have listed, and while it is more powerful than the other classes I have mentioned here, it’s probably close enough so as not to cause problems.

Instead of sorcerer, consider the dragonfire adept

Sorcerers, in 3.5, are thought to base their magic powers on trace draconic ancestry. The dragonfire adept from Dragon Magic, on the other hand, obviously gets their power from dragons. They gain a breath weapon and a number of invocations, which are at-will spell-like abilities. Similar style to a sorcerer, at a lower power level. Like the warlock, which I recommend as a replacement for wizard, below, dragonfire adepts tend to be a little on the low side, in terms of invocations known. Allowing them to learn a new invocation every level except those in which they gain a new breath effect is a simple solution that I have found works well.

Instead of wizard...

Instead of enchanter or illusionist, consider the beguiler

Beguilers, from Player’s Handbook II, get a small, specific spell list, tailored towards the casting of enchantments and illusions. They do get some opportunities to add to their list, but few enough to keep them under control. They’re also a lot of fun.

The beguiler might be the most powerful class here (I don't have enough experience with shugenjas to say for sure), so you may want to reduce it somewhat. Their armored spellcasting is a minor boon they can easily afford to lose, and as an Int-based class, 6+Int skill points is quite a bit – 4+Int might be more appropriate. Removing some or all of glitterdust, haste, solid fog, and/or time stop from their spell list might be good choices.

Instead of necromancer, consider the dread necromancer

Heroes of Horror’s dread necromancer is actually better than the wizard-necromancer at raising undead minions, but the dread necromancer spell list is tightly limited, like the beguiler’s, to enforce the focus on necromancy. Again, smaller list with carefully-controlled opportunities to expand it do a lot to limit the power of the wizard.

As will become a theme, changing the dread necromancer’s spellcasting to Intelligence rather than Charisma will make them more “wizardly.” The split between Intelligence for spells and Charisma for Rebuke Undead is also probably a good thing, and will slightly (but noticeably) play up the more “wizardly” necromancy of curses and debuffs, rather than the minion-master undead-army-raising that you get with Rebuke Undead, which I consider a good thing.

I’d also consider losing the ability to cast in light armor. It’s not especially powerful (it is easy to avoid Arcane Spell Failure), but the dread necromancer is on the higher-end of classes presented here, so chipping away at that is desirable. Plus, I recommended removing it from the beguiler (enchanter) and warlock (generalist wizard), leaving the dragonfire adept (sorcerer) and warmage (evoker) the only ones who have it: that seems appropriate.

Instead of evoker, consider the warmage

The warmage from Complete Arcane is the evocation equivalent of the beguiler or dread necromancer. Unfortunately, being blasting-only does leave the warmage a little flat, but they are still capable enough for keeping up with my other suggestions.

Since you want the warmage to be replacing an evoker-wizard, I would switch the spellcasting to Intelligence. This has the added bonus of making the Warmage Edge feature more useful (usually, outside of very-low levels, the bonus it provides is too small to justify investing much in Intelligence, and it just ends up being a trap, tricking players into putting too much into Intelligence).

Instead of wizard, consider the warlock

Another entry from Complete Arcane, the warlock is similar to the dragonfire adept, but much more strongly arcane, with abilities to conjure, dispel, divine, and transmute, not to mention their iconic eldritch blast. These make decent generalists; a bit on the weak side, so consider increasing their number of invocations known. I find having 1/level goes a long way. Also, eldritch blast goes up 1d6 every odd level... until 13th level, where it goes up every three levels. No one knows why it slows down like that, but it’s pretty common to just ignore that and use 1d6 per odd level, and that works well.

Making the warlock intelligence-based would also be a nice improvement of his wizardly-ness. It also helps the skill situation, though even with an Intelligence focus I’d still give the poor guy 4+Int skills. Losing the ability to invoke in light armor may be appropriate, especially if this justifies some more invocations known (say, 1/level, plus an additional 1 each time you reach a new tier of invocation? That would be awesome).