Around Richmond/Teddington (south west) is where the pros base themselves. Loads of green space, hills and the Thames towpath to run for miles on. Clubs: Belgrave Harriers, Hare & Hounds
Around Camden (north of the city) for Hampstead Heath/Regent's Park and the canals is another good spot. Clubs: Highgate Harriers
Around Greenwich (south east) has Greenwich Park, the Common, Oxleas and Eltham Woods and the Thames towpath. Clubs: Cambridge Harriers, Blackheath & Bromley-ish.
Centrally,a club called Serpentine are huge. Based near Hyde Park but have training sessions all over the city.
Those are the ones I have found anyway. Sure there are more.
There are plenty of a good places for running if you get out to zone 2 & 3. And from there you are a reasonable running commute from any work place in zone one.