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hippy-
duct tapes GPS to head
Garmin Forerunner watches work well for open-water swimming. Except when they very occasionally Garmin.
Other option is sticking a cycling GPS in a tow float (which are usually recommended for open-water swimming both for visibility and life-saving benefits).
FWIW I've swum in lots of random places but, right now, I wouldn't go in the tidal Thames. I've swum in the non-tidal sections (past Teddington Lock) a few times with no problems.
Greenbank
You can row, there - there are loads of boathouses by Putney Bridge. I've rowed from one years ago.
I also know someone who swam across it for a dare by Putney Bridge.