Got to confess that I have no up to date knowledge of Thames fisheries.
Seem to recall that the London Angling Association had a lot, if not most of the private fishing, also the Civil Service Angling Society. They've got Windsor racecourse fishery, Datchet as far as I know but that's been somewhat compromised by our travelling friends.
Also Reading and District had a few Thames stretches. Worth checking out.
Memories coming back about a stretch upstream of a bridge at something Clinton.
Could be wrong but it was a reputably good barbel stretch. Caught possibly my only Thames Barbel there and a 5lb+ Chub on a 5"toby in a weir pool.
Check out the CSAS website there may be more and you will certainly have relatively undisturbed fishing opportunities.
As a kid I fished with my Dad at Hurley, Wallingford and Shillingford, think they were LAA but don't ever recall being asked for a day ticket.
Have had some fantastic fishing on the Thames, mostly in weir pools, from boats, in otherwise inaccessible areas. Marlow and Goring particularly.
There's a couple of miles downstream from Teddington weir to Eel pie island that's virtually unfished due to the lack of access. There are Barbel and huge Carp there.