Beastmaster is a bit of a trap
The Hunter Archetype incorporates both the Hordebreaker and Giantslayer archetypes from the playtest into itself, letting players pick and choose their features as they level. Either way you build is a strong choice and the Hunter gets Volley or Whirlwind at lvl 11 which really ups their ability to deal damage to multiple targets in a round. In comparison the Beast Master is only getting a CR 1/4 ally which costs an action to attack. You are able to make 1 weapon attack yourself when your beast attacks at lvl 5 so at least you aren't missing out on the extra attack, but this will scale very poorly as you always only have a CR 1/4 beast as your companion. Even though the Beast eventually gets to make two attacks itself, and gains the benefit of any spells targeted at you as buffs it still is a very, very poor choice in comparison to the Hunter's abilities. Also, why would you ever want to be like Drizzt?
Ranger spell-casting is utility focused, not inherently weaker
Rangers get a lot of healing spells in their list and share them with clerics. In addition to that, Rangers have access to a lot of buffing or debuffing spells with which to help the party and impede their foes. Additionally, they have some very strong story and skill oriented spells allowing them to track and find objects and people and manipulate nature to their advantage. This synergies with the class features the core Ranger gets with their Natural Explorer and Favored enemy features.
Rangers are still a strong choice — they just aren't as purely combat oriented as they have been in previous editions
Rangers in 4e were the striker with twin-strike at level one and a plethora of nova-round dailies and encounter powers. 5e's Rangers, while very competent in combat, are designed as a class to be tied to a certain type of terrain(s) and enemy type requiring some work and buy-in from the GM and the table to reach their peak effectiveness. From a flavor perspective, the ranger is closer to the Rangers in Tolkien's works.
Like Faramir, Aragorn or even Legolas, the ranger is extremely competent in his terrain of expertise and hunting their favored enemies. In terms of a campaign, the ranger could make all the difference between barely surviving and constantly being beset by monsters while traveling the wilderness and actually thriving, ambushing enemies first, avoiding them at will, and always being able to find and forage food to keep moving.
A More Gradual Power Curve
In 2e (and 1e and Basic), though it's still a thing (by design) that fighter types are more powerful early in the game and wizards more powerful late in the game, it's less of a dramatic gap between the two because the power curves are more gradual in general. Similarly, the difference between levels isn't as extreme (a level 7 character in 2e isn't as much way-better than a level 5 as in later editions, which is why level-mixing was accepted practice then and decried as "unfair fun-killing" in 3e+).
XP Tables
The different XP progression tables mean that those other classes progress a little more quickly but the conclusion is a little more surprising. At wizard level 15, which requires 1.875M XP, the fighter is level 15 at 1.75M, but he's still a ways away from the 2M required for next - a priest is level 16 and a rogue is level 18 though! So those XP tables are used to normalize power, but the assumption is that the fighter is a lot closer to the wizard in power than those coming from later editions might assume.
Class Disparity
The same dynamic holds between classes - at early levels, a fighter has a lot more hit points and can hit things decently hard. The wizards are a LOT weaker - d4 HD, none of this "4 at first level" softball stuff. None of these additional powers to "make the wizard not feel bad when his spells are gone." And no crossbows for you. If you're a first level wizard, you cast your one magic missile and then you hide your 2 hp butt with your knife behind everyone else. They have proportionately a lot less power.
Even at higher levels, when they can do earthshaking things, they have fewer spells (and hit points, and powers, and magic items) than they do in later editions so the "my sword works all day" feature of the martials is more valuable; also they are more dependent on other party members due to the lack of Concentration and other special abilities that make them more able to independently kill. So it's still linear/geometric growth (the commonly-quoted "linear vs quadratic" premise is really a misuse of the term quadratic, which implies a parabola), but a much more restrained slope on those curves, with a lot less effective difference and a lot longer wait to be "way better than a fighter".
Being a Wizard is Hard
Fighters can just cut you. In all that lovely 1e/2e magical lore, there's a lot more restrictions on wizards. Spell components were definitely a thing. You had fewer spells and had to find more, you couldn't just make them up when you leveled. Magic item creation was pretty much totally infeasible. Things were a little more Gandalf (the article Gandalf Was A Fifth Level Magic User might as well be rewritten in 4e as "Gandalf Was A First Level Anything, Or Maybe a Commoner In The Forgotten Realms"). Many of these restrictions were removed either by rule (free spells on level) or by convention (spell components, how un-fun!) in later editions. If a wizard doesn't have to spend a lot of their time and effort scraping together magical power to get by, and can just do whatever whenever, then sure they get more powerful.
With no feats or Concentration checks or abilities to pump up saves or magic resistance, the wizard's spells just plain failed a lot more of the time. This played into the general "made do with what we have" exploratory nature of the game then; mundane equipment, player tactics, and other personal cleverness were a much larger part of play than "PRESS MY KILL POWER BUTTON".
Conclusion
3e tried to power up everyone by multiplying each class' power by 2, but that causes a much sharper gap between the class' power curves because of MATH. A wizard in 1e/2e might be able to bend the laws of space, time, and nature, but he's pretty vulnerble otherwise and really needs a team with him (until he gets to be like Mordenkainen level).
Best Answer
Ok, as you've focused down the question, let's look at class by class balance. As you stated in the question, Wizards and Sorcerers are obviously useless now--and classes that already had no or only sub-class level access are largely unaffected.
But here's the core problem: without access to magic, the rest of your formerly-magical classes must now compete with classes that were designed from the ground up to function without magic. In short, you are removing a major class feature and expecting these characters to still perform like a full class. They won't.
There's a reason that classes tend to be defined as "Non-caster, Half-caster, Full-caster." Spellcasting is not a minor component of these classes; it is pivotal to their function.
Balancing this anyway...
If you wish to proceed, you will need to accept a few things as almost certain fact.
First, your players will almost certainly not play any class that you removed spellcasting from. Your party will consist of Fighters, Monks, Rogues, and Barbarians.
Even for players who won't go down to brass tacks analyzing balance and functionality--if you tell them "you can play any class you'd like, but Spellcasting or Pact Magic is removed from your class," they're going to pick classes that don't have those features. I mean...who wants to play a class that had one of its defining ability sets ripped out.
Second, unless you go into some of the suggestions I offer below, your party will be extremely vulnerable to being overrun by mobs. The vast, vast bulk of AoE capability in D&D comes from magic. Without Area of Effect abilities it is very difficult to clear large groups of weak enemies. At the least, ensure your party has access to choke points or other means to reduce the risk of getting swamped.
Experiential Suggestions
All of this to say, I have once run a 'non-magic' game in 5th Edition. However, I did it differently than you're suggesting.
With this set up, the game worked. My players did, however, miss having access to magic and we did not turn this game into a long-running campaign.
Honestly, that first bullet point is my strongest suggestion to you. Don't offer your players gutted classes. If you're going to ban spellcasting, just ban spellcasters.
Class by Class...
Let's break it down. At the start of each class, I'll note how many levels this class now has where it gets nothing because its class feature would be tied to spellcasting.
Artificer
Dead Levels: 3 (4 if you don't consider Magical Tinkering to be useful enough)
As a half-caster who can create magic items, the Artificer is probably the most magical thing left to the game--and if you're going for a minimal-magic feel, they don't really fit the tone of the world.
That said, the Alchemist and Artillerist take a hefty nerf as their key feature requires a spell slot to use more than once a day...and the Battle Smith has lost the ability to repair their Construct.
Bard
Dead Levels: 3
The Bard can kind of function without magic, but not terribly well. They are full-casters so the bulk of their functionality is tied up in their spellcasting. What you ultimately end up with is a skill-monkey that is better at unskilled checks than a Rogue, can pass out a few buffs, and is mediocre-at-best in a fight.
Even the more martial subclasses (like Valor or Swords) don't hold up as a straight combatant when compared to pure-martials.
Cleric
Dead Levels: 5
A Cleric without Spellcasting is able to turn/destroy Undead, but is otherwise nothing but its subclass. So you essentially have 5 levels worth of class features to try to compete with all the class features in a purely martial class.
This is further exacerbated by the fact that most subclasses have features that tie into spellcasting, or are limited by how often they can Channel Divinity (twice per short/long rest). By your subclass choice, you'd end up with something like a Life Domain (so you have some healing to work with) but are limited to simple weapons and medium armor or you go War Domain and are a very-very discount Paladin.
Druid
Dead Levels: 9
Druids are the next best thing to useless...either you get a low-powered wild shape and a whopping 4 levels worth of class features (many of which are tied to spellcasting, and thus doubly useless), or you go Moon Druid and can be kinda effective as an animal, but totally useless if you get knocked out of Wild Shape or run out of uses.
Also of important note: a Druid that cannot cast spells has no features at all at level 1 besides an extra language.
Paladin
Dead Levels: 3
As a half-caster, Paladins suffer from losing spellcasting less than some of the others. But they lost the feature that most makes them powerful: Divine Smite.
What's left is a discount Fighter who generates some buff auras. However, of these 'nerfed' classes...Paladins remain useful for precisely one reason: Lay On Hands. They are one of the few options that remain that offer on-demand healing.
Ranger
Dead Levels: 3
Of the spellcasting classes, Rangers probably suffer the least...but it's still enough to cripple their utility compared to other classes. They are left with less skill utility than a Rogue, less combat ability than a Fighter, and no spellcasting to make up for their lack.
Warlock
Dead Levels: 8
Most Warlock builds are primary spellcasters and are thus now useless. You're essentially left with the Hexblade as the only thing that's still useful. But...a Hexblade with no spellcasting is little more than a crappy Fighter that can use Cha for their attack rolls.
Other noteworthy callouts
The usefulness of sub-classes like the Way of the Elements monk is not really improved by this. From my interactions with players and looking at the class itself, its main problem is that it is an inefficient sub-class. It burns Ki Points at a much higher rate than other Monks, for less effect.
Removing spellcasters from the game may cause players to feel like they have to play a subclass like Way of the Elements, just so the party isn't completely without area of effect attacks...but that doesn't mean the subclass is actually good now. Just that the players are desperate.
There is, however, a subclass whose comparative utility has skyrocketed--the Way of Mercy Monk. Of everything that is left, this is the class/subclass that has the most reliable and most frequently usable healing without accepting the burden of playing a class missing a core feature.