Much of the rural Thames path is footpath. There’s a bridlepath section from Goring to Pangbourne but after that it’s footpath again, although there are parallel bridlepaths and other cycle trails that track the river.
to be fair no-one will argue with him if he's carrying a machete
....although I have been approached by a lock key wielding septuagenarian yelling "dismount, dismount" on a tow path that was cycling allowed. the conversation went something like:
"you do realise were allowed to cycle on here?"
"yes but you putting my life in danger"
"er you're the one waving a heavy metal lock key around..."