7000ft is a lot of runway, what the heck are you flying?
So, places you might try not
listed in the AIC ?
East Midlands (EGNX) is fairly well equipped and user friendly, and probably an hour by train or two by car from the centre of London.
Bristol (EGGD) is slightly shorter than you'd prefer but similarly placed for London.
Exeter (EGTE) is about the same length as Bristol, no problem to London by train, but a dreadful journey by car.
Frankly that's about it. Anything else I can think of is either 3+ hours travel from London, or military, or in the AIC.
G