Hi,
If you are based in the South East, you can't get much better than Mildenhall and Lakenheath for military stuff, and the F-15s at Lakenheath can be particularly photogenic.
The following link will take you to a website that has all the best viewing spots:-
http://www.thunder-and-lightnings.co.uk/spotting/index.php
Just a word of warning - military airfields can be very quiet (with most closing down at the weekend), so it is best to go in the week if you can.
The only other military airfields near the South East are Northolt, Brize Norton and Boscombe Down. Both Northolt and Brize tend (like Mildenhall) to just be for transport aircraft, so if you don't have any interest in the aircraft per se you would probably be better off going to a civillian airport where the traffic is going to be much busier and the photographic opportunities much greater. Indeed, Northolt specialises in private jet movements (both civilian and military), and a trip to Luton or Farnborough will yield both more aircraft and better places to photograph them from.
You can get a good view of Boscombe Down, albeit you are not allowed to take photos looking into the base although there are no problems taking photos of aircraft on approach. The base is used for test flying, so you are more likely to see some fast jets, but again it isn't busy.
You asked about London City. There is no official viewing area, but if you take the DLR to the Prince Albert stop (rather than the airport) you can get an excellent view across the dock to the runway and apron, along with seats if you are there for a while.
Time your visit carefully though - while the airport is busy and well worth a visit from 0700-0930 and 1700-2000 outside these periods it can be very quiet. This is typical of the photos you will be able to get - the Fokker is on the runway:-
OO-VLP by
IndiaEcho Photography, on Flickr
You can see more photos from there on my Flickr pages.
Hope this helps.
Neil