A situation occurred where a cheeky Bard spoke afoul one too many times in the presence of some seriously bad hombres, and as a result, he got half his tongue cut out.

Now I'm not familiar with how much having a partial tongue interferes with speaking, but the table argued that he can say some words with great difficulty.

This thread in particular that goes into the details of casting while underwater concludes that you ultimately can, RAW, cast spells underwater, arguing that the sound produced in your larynx is the same sound produced above water, it just becomes garbled as soon as it hits the water.

Moreover, here's what the PHB says about verbal components:

Most spells require the chanting of mystic words. The words themselves aren’t the source of the spell’s power; rather, the particular combination of sounds, with specific pitch and resonance, sets the threads of magic in motion. Thus, a character who is gagged or in an area of silence, such as one created by the silence spell, can’t cast a spell with a verbal component.

So, we need Mystic Words to be spoken. Does pronunciation affect the casting? How much does intelligibility affect the casting? To extrapolate this, do characters with poor accents suffer in a similar regard if our bard cannot cast verbally due to his terrible pronunciation? The thread above dictated that you don't even need to be understood, simply the correct pitch and resonance needs to be met.

Is the bard doomed or can he mumble his way around the battlefield?

Ask your DM

Unfortunately, there really isn't any RAW for how to handle damaged tongues or even being mute. For the most part, and except for an optional damage rule from the DMG, there really aren't RAW mechanics for PCs losing body parts.

And all of this is dependent on how a DM defines the disability.

If they rule that the character is mute, then they are effectively under the Silence spell. If it is somewhere in between mute and full speech, then they'll have to make a ruling on how that will affect the Bard.

Workarounds

If the DM does rule that the character is either a mute or their speech is damaged enough to affect Verbal components, then there are some remaining options.

Cast Regenerate

This is the most obvious solution. Either have a party member or a hired caster cast Regenerate to regrow that tongue. Once regrown, all is back to normal.

Dip Sorcerer for Subtle Spell

If regenerating is not an option and they'd like to be able to cast some verbal component spells, then dipping into sorcerer would allow this. However, the downside is needing to forego bard progression in order to do this and this will require a minimum 3 level dip in order to get the Subtle Spell metamagic that would allow bypassing of verbal components.

Simply avoid verbal component spells

This is by far the least attractive option because there are only a few spells under the entire bard list that don't require a verbal component.

This happened at my table, here's what we did and how the players reacted:

In one game I was playing in, this happened to one of the PCs. Our DM ruled that they could not cast a spell with verbal components and they suffered this until we were able to regenerate their tongue.

From a player perspective, the 'maiming' of a PC was awful. It was awful when/why it happened and it was awful for them to have to deal with it for a day until our other cleric could prepare Regenerate. If you are the DM (or anyone reading who is a DM), I would be very wary about mutilating your players and giving them negative mechanical affects. It removes a lot of fun and adds a lot of table issues that can otherwise be avoided.

I think it's DM's call

Long vowel sounds and a few consonants do not require much (if any) lingual movement: B,F,H,K,M,P,R,V,W and so could conceivably be uttered without being hindered by a partially missing tongue.

Since there are no rules governing the actual words for the spells and the caster has free mouth movement otherwise, a case could be made that some spells would be entirely unaffected by the half-missing tongue, others minimally so. Which ones would be a DM call.

Additionally, the rule states "specific pitch and resonance," and a tongue definitely affects resonance. However, not all mouths/tongues are alike and the fact that there are many casters, each with a unique voice and aural characteristics, suggests that a caster who has a mutilated or missing tongue could/should relearn how to form the necessary sounds for producing the magical energy given their new oral arrangement.

Ultimately it's up to the DM, but it should carry some consequences other than making the character useless. Something similar to wild magic could be employed. It makes the character weaker in some aspects, but the character is not severely held back until regeneration is available. The implementation can vary from occurring on certain spells only, stopping after several successive successes, occurring on all spells, occurring on a cast by cast basis with no remedy other than regeneration, etc.

Depending on the type of character being played, the wild magic table could be altered to make it more applicable. If the character is particularly stealthy and avoids combat, the spells could range more into that type of magic. The backfires could prevent magic stealth and disguise from working properly, but there wouldn't be a punishment like fireball on self.

It really depends on how far you'd like to punish the player for doing something that they shouldn't have, but keep in mind that it will not be enjoyable for a character to lose all of their powers or be limited in a way that is a direct debuff instead of a workaround.

As a GM, I would rule that this situation would prevent the character from casting spells with verbal components. Frankly, this seems obvious and uncontroversial. If you didn't want to give them this limitation, you shouldn't have had someone cut their tongue in half.

Regarding the linked answer about casting underwater, I would say the difference is that in that case, the character is still able to pronounce the words, regardless of whether anyone can perceive them. In the case of your bard, the character is unable to properly pronounce the magic words, and this in my opinion renders them unable to cast spells with a verbal component.

As mentioned in other answers, there are various avenues the character can pursue to undo the damage or mitigate its effects.

On RAW part, I have nothing to add to answer by NautArch.

At my table, I would use and abuse the description of verbal component itself:

Verbal (V)

Most spells require the chanting of mystic words. The words themselves aren’t the source of the spell’s power; rather, the particular combination of sounds, with specific pitch and resonance, sets the threads of magic in motion. Thus, a character who is gagged or in an area of Silence, such as one created by the Silence spell, can’t Cast a Spell with a verbal component.

Emphasis mine. Thus, I would allow Bard to use his spellcasting focus instrument in place of actual words, assuming he is sufficiently good at getting specific pitch and resonance. After all, Wizard who can't even sing Mary Had a Little Lamb in tune can do it with his voice, why wouldn't maestro be able to do it with his instrument?

I tried it in earlier edition. In 3.5 description of verbal component was way less forgiving, but once upon a time we tried it. Bard wasn't able to cast anything language-dependent, of course. It was a hindrance enough to teach him a lesson, but it didn't render him useless. Still, that was hindrance enough that pretty soon rules and events was bent and ignored to allow him to heal and talk normally, if maybe little funny (fluff, no mechanical effect).

It's basically DMs wits against the players. Don't be shy about a hard campaign, because the DM has to be equally sophisticated and they'll probably tire if they can't do it either. So keep all options open in gameplay, explore dialog (which will test the DMs wits especially), and try your luck a few times.