Plenty of Moorings at Richmond, Windsor, Shepperton, Henley, Goring, Abingdon and more than enough past Osney as most cruisers can’t get under the bridge. You can also moor at many rural places such as fields etc. If leaving the boat I’d phone a few lock keepers as many have mooring available or leave it at a marina. You may/will have to pay at many of these places.
As for anchoring in a river overnight , Hmm not sure I’d advise of that. You need to display an anchor light if you do and you may still fall foul of the EA