CMOT, I think the Wallingford site mentioned was Bridge Villa - on the opposite side of the road from the paddling pool (as was! ) and the smaller campsite beyond that. As far as I know (my PIL use it), Bridge Farm has no access probs beyond crossing the road to get to the footpath side of the bridge over the river. And has EHU. 
IME, it has to flood seriously to waterlog Bridge Villa campsite. The field (water meadow) beyond does get very swampy but the campsite is normally fine.
White Mark Farm is fab! Watlington is a bus ride away from Oxford, or there's park and ride.
The CCC one behind Go Outdoors is bang next to Redbridge Park and Ride, which is handy if you think you'll be going into Oxford a lot, but has train lines just behind, and also some sort of scrapyard (?) or something - anyway, something bloody noisy with lots of mechanical din. Handy if you run out of gas, though. 
Oxfordshire is lovely when it stops raining and there are lots of 'pottering' things to do in that particular area, South Oxfordshire.