If you're limited to public transport, within London,
- RAF Museum, Hendon
- Imperial War Museum, London
- Cabinet War Rooms
- Science Museum, aviation parts.
First's free, second and third are the same organisation - take out a year's membership and it'll pay for itself quickly enough. Fourth is also free. On which point, if you are able to grab a car, go North a little and try to do one or both of...
- Imperial War Museum, Duxford
- Shuttleworth Collection, Old Warden.
Think of Duxford as a British Chino, it's very much one of the world's greatest aviation museums. It also incorporates the American Aviation Museum, which is impressive, but a load of stuff most Americans haven't had a chance to see as well.
Shuttleworth has a mindblowing collection of early aeroplanes, cars and motorbikes, a very large proportion in flying condition. They also do open workshop days between Christmas and New Year - check their website. I can't think of anywhere in the world similar.
Duxford's a (very) full day, Shuttleworth maybe half a day when things aren't flying. They're not far from each other.
Heading West, just about do-able by public transport you might also consider if it's a nice day, West London Aero Club at White Waltham. Huge amounts of history, and one of the biggest flying clubs in the UK, with a selection of fun aeroplanes if you don't mind paying (or catch one of us members when we're around and free). Aeroplanes from Super Cubs to a 2-seat Hurricane are available, although the latter is understandably very expensive.
G