It will take some work, but it's perfectly doable.

Enemies with special defenses

The players won't have access to see invisible, flight, magic weapons, or ghost touch. Therefore, if you include enemies that can fly, or have DR/magic, or are intangible, these will be huge challenges, and may be downright impossible. The easy solution is simply to not include such enemies, but the more interesting way is to treat them as nigh-invulnerable enemies that the heroes will have to figure out how to defeat. In a normal Pathfinder game, a ghost is a normal critter with a +2 CR template pasted on top, and requires some minimal preparation to defeat. In a low-magic Pathfinder game, a ghost is a mystery: why did they become a ghost? How can you persuade them to pass on, or at least let the party pass by peacefully? Can you persuade the local priest to perform an exorcism, and will it even work? Instead of "find monster, insert fireball," these types of encounters are now role-playing challenges, because they can't be solved any other way!

Alternatively, you can simply strip out the special defenses from enemies. Pathfinder assumes you have level-appropriate counters to special abilities anyway, so by removing those special defenses, you aren't going too far from the original intent. Adjusting the CR is left as an exercise for the GM, because it's going to take a fair amount of trial-and-error to determine what the right balance is.

Fixing armor class, and other numbers issues

Pathfinder assumes that both attack bonuses and defenses will be augmented by magic items. This partially balances out if the players don't have magic items, but consider giving everyone a +1 bonus to AC and all defenses every four levels. Don't make them pay a feat for it, just give it to them.

While you're at it, give your players bonus XP for the monsters they defeat, by calculating the XP as if the monsters were a higher CR. Since they're operating without magic, every encounter is going to be harder than what the DMG "expects" when it calculates XP per CR.

What will they do with their money?

Your players won't be able to buy gear that personally enhances their ability to make things dead faster, or grant them new solutions. If you keep to the normal loot rules, then the party will have far, far more money than they know what to do with. You have two options here: give them less money, or give them something to do with that money.

Let them invest in mercenary companies or land holdings. Let them become influential in the church, or their hometown, or even their country as their economic might and donations in the right places give them power that they would never be able to take with a sword. Favors in high places give characters some very powerful options.

Recovery after combat

This will require explicit house rules; you'll need to accelerate natural healing (heal a percentage of HP per day instead of a flat amount?), allow Heal checks to do much more than they normally do, grant the local clergy some extremely localized healing powers (they can heal people brought to their church, but not outside of their place of worship), and/or make this a political game rather than a hack-and-slash game.

If everyone's having fun, then it's a good game. It doesn't matter if the characters aren't optimized: as long as they feel like they're making a difference in the world and they're enjoying the game, then you're doing it right. The characters will be balanced, more or less: they all don't have access to magic, so intra-party balance isn't as much of a problem. You'll see that the players lack all of the magic-based solutions that you'd expect in a normal Pathfinder game, and you'll select the enemies more carefully, but things will work out fine. Let your players know that things will be a bit different than normal, and your players will go along with it; they requested this kind of game, after all.

Arcanist, Sorcerer, and Wizard are three of the most dominatingly-powerful classes in the game

Each of these classes, built right, can do just about anything, and in many cases can do very close to everything. The spell list they have access to is the best in the game, and spellcasting is the best class feature in the game.

These three can nullify a huge array of problems with a standard action. They are all extremely capable of getting new solutions in those cases they run into a problem they cannot already solve.

At very-low levels, these classes have somewhat-low stamina, and their spells are not nearly as dominant as they quickly become, but smart and careful play can ensure statistical advantages in almost every endeavor. As they level up, their stamina and power increases exponentially, and it is not long before they are clearly superior to almost everything else. Around 7th-level, at the latest.

This does, however, rely on careful and creative play. Extremely thorough knowledge of the massive sorcerer/wizard spell list, and what spells are best for which situations, is required. It takes a lot of work and mastery to accomplish. Just as the ceiling for these classes is very high, rock bottom is a very dark and dank bottom indeed. That said, they have exceptional ability to fix any mistakes they make; even if they play very poorly for a long time, they can very suddenly become powerhouses by simply preparing better spells (the sorcerer is, obviously, somewhat worse off in this regard).

Summoner and witch are close

The summoner and witch are very close in power to the three listed above. The eidolon is an incredibly versatile and powerful class feature, summon monster offers a great deal of utility, and the witch is pretty much a lesser wizard.

The summoner has the side-benefit of being much easier; it’s pretty easy (and effective) to hide behind the eidolon. The witch is actually kind of harder, though probably more powerful; the familiar is a massive, glaring vulnerability, and the spell list just isn’t quite what the sor/wiz one is. But then, the witch gets to heal status ailments, which none of these classes get, and though it’s supremely limited, that could be a big deal (see below).

Alchemist, bard, and magus are more well-rounded

These classes are not nearly as powerful as the above, but like the summoner they can be a bit easier, particularly the magus, as they are sturdier before spells come into play. In particular, they also allow for more up-front characters, which is not only something many enjoy, but also is often a useful platform for the more magely casters to put buffs onto.

These classes are reasonably competent, and are in a pretty sweet spot, power-wise. They also add a bit of variety compared to the above (though the summoner’s eidolon does a depressingly good job filling the melee role). A group at this tier, or at this tier with support from the above, will be very competent.

The alchemist is also very important because it is the only class here that gets lesser restoration and restoration, which are otherwise major gaps in the capabilities of arcane spellcasting.

I have no experience with bloodrager, investigator, or skald

Just to be upfront about it: I have no practical experience with these classes, unlike the above. So take the below with a grain of salt.

That said, the bloodrager’s spellcasting is extremely weak, and nerfed rage powers are not a particularly exciting thing to get in return. This class looks like it could be quite fun in a low-power game, but I would not recommend one in a game alongside the above.

Skald looks better; its spellcasting is closer to the bard’s, and it gets enough barbarian powers to be interesting. I actually want to see this class in play at some point. Probably plays well at around the bard tier?

The investigator seems to be on a similar page as the skald, though the existence of the vivisectionist alchemist makes me wonder what the point of it is. I rather like the concept, anyway.

One major thing arcana is lacking: healing things other than HP

HP healing won’t really be a problem; even the bard can handle that well enough with a wand of cure light wounds (though make sure the party does get such a wand somehow), and the witch is actually pretty good at it, but healing status effects, ability damage, and conditions is going to be much harder. The cleric list is pretty much the only places you find really good healing of that sort.

The alchemist and witch are the stars here: the alchemist gets the crucial lesser restoration and restoration spells (though he needs the Infuse Extract discovery to share them), and the witch gets the various remove ailment spells, plus heal at very-high levels. Between them, they cover the major healing options that are otherwise missing from arcane spellcasting. The cleric still has major advantages (not least of which being that the cleric can handle both of these things), but if both of these classes are represented in the party, they can probably handle these things about as well.

If either or both of these classes does not get chosen by players, be aware that anything that inflicts an ongoing status effect is likely to force the party to retreat, to find a temple to get it dealt with. That, or if absolutely forced, to throw themselves into later encounters at a massive disadvantage. Deciding how you want the campaign to play out with this reality would be really important. Items and institutional resources are likely going to be necessary, which means you, as DM, have to make them available.