Wow. I guess, with a puke being no more than a half a litre, and all the chunks well filtered, and a children's teaching pool containing approximately 150,000 litres (I have assumed a length of 15m, depth of 1m, width of 10m), then once filtered, there should be a dilution of one part puke to 300,000 parts water. Hardly a health issue, in heavily chlorinated water.
In my DS's teaching pool, one child pissed in the pool (as if they all don't, ffs), all lessons were postponed until they had drained the pool, cleaned it with disinfectants and refilled it. A horrendous waste of water and other resources (including parent's time) for a completely benign and miniscule contamination.
People really worry about this sort of calamity too much, methinks. We're almost all happy to swim in the sea, which is used as a toilet by every fish, whale, and turtle, and by all humans living within earshot of the surf. Perhaps OP ought to listen to DP a bit more in order to become more relaxed.