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.
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.
- Spellcasting classes were simply forbidden. Rather than leaving gutted classes available for someone to pick and end up miserable playing, I just banned any class that used magic.
- Go homebrew digging. In order to compensate for the lack of magic, I went digging for homebrew content that could fill in the lack. Ultimately, I found a Warlord and a Psion class that I was happy with, providing a 'martial healer' and 'non-magic spellcaster' option so the party could still have main-class access to healing and area-effect abilities. There are also some decent "Non-magic Ranger" homebrews out there.
- Avoid mobs: I almost wiped my party with a swarm of Blights--but this is the case any time you have a party short on AoE ability.
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.
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.
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.
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.
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.
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.
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.
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.