With the exception of multiclassing, is there any big reason that if you were to change the spellcasting ability of a (spellcasting) class it would be a bad idea?
As a specific example: Let's say I was switching Sorcerers from Cha to Wis, what unexpected effects could this have on that class's balance? Additionally let's assume all places in spellcasting where you added your charisma modifier instead you now add wisdom (and vice-versa).
This is the wonder of RPGs, they are flexible. But they are not without their trade-offs here.
Probably not an issue
Probable issues
Let's break these out.
Physical vs Mental
Casting is generally tied to one of the three Mental stats (Int, Wis, Cha). If you allowed an Eldritch Knight Fighter to make their casting stat Str this would probably cause power issues because you would be letting them "double-down" on their best stats. Likewise for Arcane Tricksters, Paladins, Rangers.
Saves
The game technically has six saves, but Dex, Con & Wis are really the most important saves. If you look at the Save Proficiencies by class, each class only gains proficiency in one of these two saves. They also typically gain proficiency in the save that matches their best stat.
So Sorcerers get proficiency in Cha saves making them really good at Cha saves. Rogues are really good a Dex saves, Clerics are really good at Wis saves, etc. If you make a Wis Sorcerer, they become "pretty good" at Cha saves, but also pretty good at Wis saves.
This definitely changes the nature of the class. You normally don't target a Cleric or Druid with a Wis save because it's probably not going to work. But now it's not so clear. This will affect both PCs and NPC casters.
Skills
Available skills tend to be connected to a character's primary stat. So Clerics get more Wis skills and Wizards get more Int skills. If you switch this around, it really reduces the odds of having someone particularly skilled at one of these disciplines. If the Cleric doesn't have a good Medicine score, then who does? Same for Arcana?
Fortunately, some of these can be made up for with specific Backgrounds, but it does hamper specialization usually connected with the given classes.
Flavour
There is a specific history behind the various casting stats. Int casters had to study, Wis casters are divinely inspired, Cha casters are "naturals". When you switch this stuff around, you're also implying a different world vision than the "classic" D&D.
This is not "bad" per se. But it is significant. Why does a low Int Wizard use a spell book? Are they really going to be book worms if they use Cha to cast? If Sorcerers use Int to cast, but don't need books is there a reason for that? If Clerics use Int to cast, then where does that whole "divine inspiration" thing kick in?
These questions can clearly be reconciled, but they are meaningful. They say something about your game world that is different from say a classic Forgotten Realms world.
Outside of skills (and potential saving throws), for a single-class character there aren't any specific consequences of choosing one attribute over another. All spells are designed around spell modifiers. Whichever attribute you use for that modifier makes no difference.
But skills are nothing to scoff at. In case of your example:
There are five Wisdom skills, but there are four Charisma skills, and Wisdom skills are in general more versatile (as the Charisma skills often end up being different ways to accomplish the same result, but not always). Presuming you'd put most of your points in Wisdom, this gives a greater advantage in a general situation. But you also have to take passive perception into account, which is almost a skill of its own. There's nothing game breaking here, but the Wisdom skills provide considerably more utility than Charisma. Experienced and/or imaginative players will know how to take advantage of this (should they choose to do so).
Though you can't Medicine your way out of combat, so if your PC's aren't a bunch of bloodthirsty savages his lower Charisma will make it harder for him to try and settle things in a peaceful manner (which, ironically, is something a wise character should be prone to trying).
In general, if you find your Sorcerer taking advantage of his Wisdom skills often, try to counter-balance it by increasing the frequency of situations where Charisma truly shines.
Switching spellcasting to a different mental stat shouldn't be a big deal. You certainly don't want to switch it to Con (or Dex ::shudder::), but you already know that.
I would rank the mental stats in the following order: Wis, Cha, Int, so I wouldn't have any problem if a player wanted his character to use Int instead of either of the other two.
I wouldn't allow every player to chose a Wisdom based Wizard for the better skills and saves, but if a player had a particular character concept in mind and Wis fit better than Int, I don't see how it would break the game.
Edit: DaFluid has a good point on multi-classing. If someone talks you into a Charisma based Wizard, then multiclasses to Warlock, you are well within your rights to call foul and insist they chose a different stat for their second class's casting.
Depends on the attribute.
As DaFluid mentioned, you could switch a Sorceror over to Wisdom from Charisma and not completely annihilate the game's rule system. Your Sorceror will be considerably better at detecting things and making Will saves, but he will also be worse at convincing people to do things. However, care has to be taken with the physical attributes, which grant innate bonuses besides skill benefit. Namely, Constitution determines hitpoints, Dexterity determines armor class, and Strength determines melee attack, melee damage, and carrying capacity.
Even "physical" casters can be balanced, but it takes work.
In the current campaign I'm playing, in a custom dying-Earth-styled setting, one of our players is using a custom race with an infatuation with eating meat of all kinds. His character relishes the opportunity to butcher a completely new creature and later sample its flesh (believe it or not, he's not evil because cannabalism is common in this setting -- he just likes meat). He is also a Wizard with 18 Constitution and, what, 7 Intelligence? He is using a custom variant called a "vivimancer" that specializes in shaping flesh and biological function, and they use Constitution as their spellcasting attribute.
What this amounts to is a Wizard that plays substantially different from normal Wizards. His spells are almost entirely utility, with some limited healing and summoning. His skill checks are frankly quite bad, as well. In return, he's pretty hard to kill, and the GM has also allowed him to eat the brains of dead casters for a chance to learn their spells (this is also due to the lack of spellbooks). The GM has a table from somewhere that he rolls on -- sometimes eating Wizard brains has quite negative effects for him. Additionally, food is rarely a problem for him because of his iron stomach. He'll eat any and every dead creature.
I would not say that his class variant is stronger than a conventional Wizard. He has less options than a Wizard does, but is also safer, from both a combat and a non-combat perspective (less prone to death or harm from the elements, better Con save, etc). He still has plenty of effective spells, including a few choice crowd-control ones. I think he is quite balanced overall. However, it required our GM to draft a completely new spell list (including some custom spells) and class variant features.
