They were watching the kid in much the same way that Morpheus and company watched Neo in the first movie. In the Kid's story, Neo calls him, and in an scene reminiscent to the movie, tells him to get out of the classroom now. The kid is chased by agents to the roof, and eventually jumps off.
The kid presumably "wakes up" from the matrix and is disposed of the same way Neo was.
Recall that in the first movie, the red pill tracer program was taken by Neo and then he is woken up less than 5 minutes later. So the ship had to have been close enough to Neo's location in the physical world to get to him relatively quickly. Perhaps the same was true for the kid. They knew something was going to go down (Neo called the kid to warn him) and when they saw him "self substantiate" (the first ever to do so) they reacted quickly and made it to the pools in time to recover him.
Maybe the kid even helped his own cause by treading water (difficult with unused muscles) or making it to the edge where he could hold on. Also, we don't know if all disposal pools are the same, so perhaps the one where the kid was dumped wasn't as deep and he didn't need to actively do anything to prevent drowning, giving the ship more time to find him.
Lastly, recall that now Morpheus and team have Neo on their side - someone who is able to intuitively interact with the machine world. Perhaps he was able to provide some insight as to where the kid was in time to rescue him.