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Walking on the Thames foreshore is allowed. Picking up things or digging requires a permit although applications are closed.
https://www.pla.co.uk/Environment/Thames-foreshore-permits
Can I visit the foreshore without a permit?
While you may visit the foreshore, you may not search the tidal Thames foreshore from Teddington to the Thames Barrier - in any way for any reason. This includes all searching, metal detecting, ‘beachcombing’, scraping and digging.
There are more details on that page including one bit to avoid because of a dangerous structure
I haven't tried yet but I think many can be got with a walk along the foreshore at a low tide.