Oh they do. The water is murky but the bottom is nice and soft to tread on.
You have to sign a disclaimer to join the club. But I think the danger to health was more so a problem last summer with an algal bloom. If that returns this summer then it might be an Olympic Triathlon issue.
Hygiene is always a risk, wherever you're wild swimming (although the Serpentine is not the wildest place but is about as wild as you can get in London, along with Hampstead ponds). Wiles disease (rat pee related I think) is maybe the best known nastiness. David Walliams seemed to get something from the Thames but was peeing willy-nilly to warm up his wetsuit during his sports relief swim.