Good advice so far.
You need at least a day with an early start for Duxford. Hendon, as Mr B has said closes the Grahme White (Great War) building at midday. However the Battle of Britain Hall is, IIRC, only open in the afternoon. A great collection, poorly managed.
A couple of alternatives if time is short is the Imperial War Museum, Lambeth or the Science Museum South Kensington. The IWM should be a bus ride for you (it about 1/2 miles south of the Thames) while South Ken isn't hard to get to and is more central than Hendon.
The Science Museum's Aeronautics gallery contains one of the best collections of original, record-breaking aircraft outside the Smithsonian:
Alcock & Brown's Vimy -
first aircraft across the Atlantic, non stop.
Supermarine S6B - Winner of the Schneider Trophy.
The only original Fokker Eindekker left.
E28/39 - Britain's first jet aircraft, the oldest surviving jet in the world.
Amy Johnson's DH 60 Jason - First woman solo pilot to Australia.
And so on. There's also a selection in the 'Making of the Modern World' and some interesting aircraft stuff in the temporary exhibition 'Dan Dare & The Making Of Modern Britain'.
The IWM has a genuine Battle of Britain Spitfire, V2, Mustang, Fw190, He 162, Lancaster and Halifax noses and numerous other items.
http://london.iwm.org.uk/
http://www.sciencemuseum.org.uk/
Route planning, hours etc here...
HTH!