The initial preposition at in such contexts is entirely optional, but it usually wouldn't be included (although in reality we usually use when rather than [at] what time anyway :).
OP's specific example happens to include a "location-based" clause based on at [the swimming pool], but it might be worth looking at two slightly different contexts...
1a: What time does the shop open?
1b: At what time does the shop open?
1c: What time does the shop open at?
2a: Where did you come?
2b: From where did you come?
2c: Where did you come from?
In my opinion, both 'b' versions above are at least slightly stilted / awkward. But whereas 1a and 1c carry the same meaning, that's not the case with the second pair. 2a (without the preposition from) is effectively asking where you ended up, not where you started from (speaker might be asking your final position in a race, for example; Where did I come [in the marathon]? I did pretty good, actually - I came third, out of 2000 runners).
The point being that because there's no credible alternative meaning in the first pair that depends on whether the preposition is included or not, it's entirely a stylistic choice (and on average we don't bother with unnecessary words). But the second example shows that we do include the preposition where it's required to avoid ambiguity.