I think that your best bet is to look to joining the
Spitfire Flying Club at Popham - admittedly a bit of a treck out of London, but in the right direction. Popham has a plethora of
variously priced and usually very friendly syndicates, a real "pro-aviation" attitude, a well regarded resident instructor when you need one, and is in a really good spot to go flying from.
There are a few reasonably priced aeroplanes to rent from private owners down there as well.
There's usually not a complete set of information about the activities and aeroplanes at Popham online - everything revolves around the clubhouse. I'd recommend going down there on a nice day for a few hours, chat to the locals, have lunch, look at the noticeboard, and work from there.
G