2 deleted 12 characters in body
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Firstly, you didn't ask, but movement is effectively cut in half, rather than quartered.

While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a swimming or climbing speed.

So movement should only be ¼is ⅓ if the heroes are in difficult terrain - normal swimming is only ½.

Next, spellcasting. There are no specific rules about this, but:

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.

It's pretty reasonable to extend this to a character underwater, so spells with verbal components are out. There doesn't seem to be anything else that would restrict underwater spellcasting, however.

Finally, breathing mechanics. The rules for this are on page 183 of the PHB, and are much simpler:

A creature can hold its breath for a number of minutes equal to 1 + its Constitution modifier (minimum of 30 seconds).

When a creature runs out of breath, it can survive for a number of rounds equal to its Constitution modifier (minimum 1 round). At the start of its next turn, it drops to 0 hit points and is dying, and it can’t regain hit points or be stabilized until it can breathe again. For example, a creature with a Constitution of 14 can hold its breath for 3 minutes. If it starts suffocating, it has 2 rounds to reach air before it drops to 0 hit points.

Firstly, you didn't ask, but movement is effectively cut in half, rather than quartered.

While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a swimming or climbing speed.

So movement should only be ¼ if the heroes are in difficult terrain - normal swimming is only ½.

Next, spellcasting. There are no specific rules about this, but:

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.

It's pretty reasonable to extend this to a character underwater, so spells with verbal components are out. There doesn't seem to be anything else that would restrict underwater spellcasting, however.

Finally, breathing mechanics. The rules for this are on page 183 of the PHB, and are much simpler:

A creature can hold its breath for a number of minutes equal to 1 + its Constitution modifier (minimum of 30 seconds).

When a creature runs out of breath, it can survive for a number of rounds equal to its Constitution modifier (minimum 1 round). At the start of its next turn, it drops to 0 hit points and is dying, and it can’t regain hit points or be stabilized until it can breathe again. For example, a creature with a Constitution of 14 can hold its breath for 3 minutes. If it starts suffocating, it has 2 rounds to reach air before it drops to 0 hit points.

Firstly, you didn't ask, but movement is effectively cut in half, rather than quartered.

While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a swimming or climbing speed.

So movement is ⅓ if the heroes are in difficult terrain - normal swimming is only ½.

Next, spellcasting. There are no specific rules about this, but:

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.

It's pretty reasonable to extend this to a character underwater, so spells with verbal components are out. There doesn't seem to be anything else that would restrict underwater spellcasting, however.

Finally, breathing mechanics. The rules for this are on page 183 of the PHB, and are much simpler:

A creature can hold its breath for a number of minutes equal to 1 + its Constitution modifier (minimum of 30 seconds).

When a creature runs out of breath, it can survive for a number of rounds equal to its Constitution modifier (minimum 1 round). At the start of its next turn, it drops to 0 hit points and is dying, and it can’t regain hit points or be stabilized until it can breathe again. For example, a creature with a Constitution of 14 can hold its breath for 3 minutes. If it starts suffocating, it has 2 rounds to reach air before it drops to 0 hit points.

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Firstly, you didn't ask, but movement is effectively cut in half, rather than quartered.

While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a swimming or climbing speed.

So movement should only be ¼ if the heroes are in difficult terrain - normal swimming is only ½.

Next, spellcasting. There are no specific rules about this, but:

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.

It's pretty reasonable to extend this to a character underwater, so spells with verbal components are out. There doesn't seem to be anything else that would restrict underwater spellcasting, however.

Finally, breathing mechanics. The rules for this are on page 183 of the PHB, and are much simpler:

A creature can hold its breath for a number of minutes equal to 1 + its Constitution modifier (minimum of 30 seconds).

When a creature runs out of breath, it can survive for a number of rounds equal to its Constitution modifier (minimum 1 round). At the start of its next turn, it drops to 0 hit points and is dying, and it can’t regain hit points or be stabilized until it can breathe again. For example, a creature with a Constitution of 14 can hold its breath for 3 minutes. If it starts suffocating, it has 2 rounds to reach air before it drops to 0 hit points.