The same thing as any other prone creature
Jeremy Crawford has confirmed the answer that follows as a correct interpretation:
If you're knocked prone underwater, you're subjected to the effects of the prone condition as normal. One way to visualize it is that you're floundering.
There are no conditions/rules/restriction in the game about when a creature can be prone. So when a game effect says that a creature is knocked prone, they follow the same rules as any other prone creature.
The effects of being prone are:
- A prone creature's only movement option is to crawl, unless it stands up and thereby ends the condition.
- The creature has disadvantage on attack rolls.
- An attack roll against the creature has advantage if the attacker is within 5 feet of the creature. Otherwise, the attack roll has
To flesh out our understanding of being prone underwater though we should look at a few helpful things:
Prone doesn't necessarily mean flat on the ground
I think it is important to note the prone is a condition in the game, but not necessarily always a physical description of the physical state of the afflicted creature. The primary purpose of the condition is to describe the mechanical effects on the character. Thus prone could mean different things in different circumstances.
I cannot find anything explicitly addressing this, but we can learn from an example that is addressed: being prone while flying.
Prone condition also exists for flying creatures
Firstly, a flying creature can be knocked prone:
If a flying creature is knocked prone... (PHB)
That makes as little or even less sense than a swimming creature being knocked prone in the common sense of the word. Yet it is explicitly possible and would entail the same mechanical effects as any other creature being prone (with the additional downside of also now falling but that is specific to flying creatures).
Secondly, according to an optional rule:
...it can halt the fall on its turn by spending half its flying speed to counter the prone condition (as if it were standing up in midair) (XGtE)
The fact that it says "as if" it were standing up tells me that the game is acknowledging that prone is not literally describing the physical state of lying flat (which as we talked about above, would be a ridiculous if not impossible state for a flying/falling creature to be in). Thus this is a mechanical effect more than a descriptor of the physical state of the character.
Swimming creatures should apply prone the same as any other creature
This applies equally well to the thought of swimming creatures. As such I see no reason why prone should be any different for swimming creatures as it is for flying creatures or creatures on the ground, even if it makes little to no sense physically.
Describing underwater prone in-fiction
If I were DMing a situation where a swimming creature was knocked prone I might describe it to the character as the creature getting flipped over to an inconvenient axis of their body and having to struggle to right themselves to function normally.