Try searching for skiffing clubs, there are a few on the Thames west of London. Skiffs have no moving seats, just fixed thwarts.
Many rowing clubs are gearing up for more recreational rowing, rather than racing, with wider, flatter-bottomed boats (as Mike describes, and also carbon / Kevlar lighter weight ones). They do have sliding seats, but are quite a bit more stable than racing shells, if more prone to skating across the river in sidewinds. I would imagine they might be a nightmare on the tidal Thames, but fine upstream (west of Teddington, I think).
We have many adults learning to row, and cater from pottering about to racing and the heady heights of national champions (we host a British Rowing 'Start'group - all huge and scarily fast). If you get going with it, that adds another potential trip from Mildenhall, to come sculling in Ely as a guest - and other YACFers can watch and shout insults from the top of the bank.