Paddington's not a bad spot to live if you like city living - I'm a country dweller myself and jump on the train if I have to go into London, but have friends in Paddington and it's not unpleasant.
By public transport, Denham (small friendly spot, but surrounded by controlled airspace), Wycombe Air Park (great airport facilities, no social side), White Waltham (great social side, lots of interesting aeroplanes, all grass runways), Stapleford, (busy training airfield), Elstree (even busier training airfield) all spring to mind for relatively easy travel by public transport. I wouldn't recommend Biggin Hill - it's very much set up for business aviation, and not an affordable place to fly privately. You might also look at Popham - further afield but easy by train from central London and a great place for friendly affordable GA. All of them you're looking at a train then taxi / folding bike / long walk, but it's do-able.
G