In no particular order: LHR, LGW, LCY, LTN, STN, SEN, LYX and OXF (Endeavour Morse International) already claim to be "London" airports.
NHT (maybe one day), FAB, MSE, SOU and BOH ("London West" in FR-speak?) probably have have a prior claim to this status ahead of BHX.
Bizarrely LHR doesn't need to use the "London" prefix any more because the name "Heathrow" is so well-known worldwide.