while the thames can be swam upriver of Putney Bridge, there are conditions as the Thames is a busy river. but there are other spots in London where one can do wild swimming.

for example:

london royal docks https://loveopenwater.co.uk/swimming-london-royal-docks/

canary wharf: https://canarywharf.com/whats-on/open-water-swimming/

the famous ponds in hampstead heath: https://www.cityoflondon.gov.uk/things-to-do/green-spaces/ha...

the famous hyde park lido: https://www.royalparks.org.uk/parks/hyde-park/things-to-see-...