TJ, don't be so Edinburgh-centric! North and South Downs has some fantastic riding on it, very varied. OK in winter it can be a claggy nightmare but then the Pentlands isn't exactly any better most of the year now is it?
Certainly from S.London you can be off-road within a couple of miles (urban parks etc) and into proper countryside outside of Croydon within 40 mins.
From central London I'd jump on a train, get off at Dorking, Reigate, Crawley, Guildford, Box Hill – endless possibilities. Or out to Martins Heron for Swinley Forest, that's an hour or so by train from Waterloo. In fact we used to do that as a nightride – get the first train out after rush hour, ride Swinley then get the last train back at about 11pm. Penshurst Off-Road Club is about 30 miles south (an hours drive), that has regular short-course DH and dual-slalom races as well as being an excellent little venue for general riding.
I did more racing and riding living in London than I've managed anywhere else – Herne Hill velodrome runs Monday and Wednesday races through summer as well as training sessions. Hillingdon Cycle Circuit, Crystal Palace Circuit and the (now gone) Eastway Circuit had races 2 or 3 weeknights and at weekends there was always a Gorrick MTB or a Surrey League road race etc. Plus the easy access to loads of MTBing.
Edinburgh has the Pentlands and a couple of other local-ish bits otherwise it's a drive out anywhere else…