you won't be able to ride from the airports to london, but kingston to central london is probably do-able in an hour and a bit. [about the same time as taking the train]
you can take bikes on overground trains out of rush hour.
you can take bikes on the tube, but not underground
and you can't take bikes on busses.
as said, brixton cycles is awesome. condor cycles on chancery lane is also good, but tend towards high end/race stuff [think corima rims, campy record].
online,
st johns street cycles are really good.
fsnl
sparky