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Yep, boats and kayaks are allowed there. I've seen people paddleboard along there too.
I'm missing about 20 rivers tiles between Kew Bridge and Tower Bridge, all but about 6 of them are easily gettable via public roads/paths. And by the time you get up towards Waterloo/Blackfriars/Southwark bridges the river is a lot busier with river traffic so these tricky 3 or 4 are about it for now.
Would all of this be easier with a boat? Are you allowed to row or kayak along the Thames there?