What part of London are you in? I’m assuming that a Lido isn’t what you’re after?
If you’re in north London, I’ve been swimming here since the season started, and it’s lovely and clear and peaceful. You have to do an induction first, but after that you can turn up on Saturday or Sunday mornings or Thursday evenings when you want:
https://www.better.org.uk/leisure-centre/london/hackney/west-reservoir-centre
I don’t really like the Serpentine (it’s really grim) or Royal Docks (not that pleasant), but Hampstead Ponds are lovely, if a bit crowded on a hot weekend. Denham Lake is meant to be good, out by Uxbridge, too.
I’m sure that there are places near Windsor/Thorpe Park that also let you swim in them, as well.