There are flying clubs at Oxford/Kidlington airport itself. There is Oxford Flyers, which is a subsidiary of OAT, but sort of separate from the College, whilst within it. But there are also other flying schools on the airport.
Additionally, there is Enstone Airfield, close to Oxford, on the Chipping Norton road. There's a flying club/rental club there as well.
Sorry if this all a bit vague. But it's enough to give you leads to check up on if you're interested enough. I fly with Oxford Flyers, myself, so I haven't investigated the others in detail.
All the best,
Paul