There’s a few effects which can murderize you via asphyxiation — the darkmantle, the water wierd, the water elemental, the animated rug — but there really aren’t rules for asphyxiation.
By the book, it’s a minute per point of con bonus before you care, and then a round per point of con mod, and then you are unconscious and dying.
By my rules, you acquire 2 temporary levels of exhaustion at the end of each minute, DC 15 constitution saving throw for half.
Both are unsatisfying to the player who has just been nobbled by a large ‘gator.
So: Normally, the “holding your breath” rules are a minute by minute thing. Any effect which would break concentration can break your breath, pushing you immediately down to a round-by-round basis (which on the normal rules gives you 1 round per point of constitution bonus before you fall unconscious, and on my rules puts you in a death spiral where you’re making constitution saves and acquiring levels of exhaustion).
So in our darkmantle encounter, let’s say the victim — er, player character — is a level 5 fighter with 50 hit points and a +3 constitution bonus. The darkmantle uses crush (whose text indicates the target cannot breathe), and hits. The unable-to-breathe fighter is now holding breath; by the book, no effects for 3 minutes, by my rules no effects for 1 minute. They try to escape at the top of their round and fail, that’s fine. The darkmantle uses crush and hits (it had advantage this time) — this triggers the fighter to make a concentration check — if the darkmantle crushed for 6 damage, the DC would be 10 (as concentration check’s DC are 10 or half damage). Fighter passes and they still have their breath; if they’d failed, they’d lose their breath and enter into (by the book) 3 rounds before unconsciousness, or by my rules, 3-6 rounds of increasing levels of exhaustion.
Garrotes, by the way, allow a humanoid to take advantage of this: they are used as a nonproficient finesse weapon attack against a grappled target against whom you have advantage (prone, restrained, blinded, …). While that target is grappled, it cannot breathe.
They explicitly target AC to allow things like a gorget to count.