Interesting thread. Some day I will take my bike overseas but haven’t got round to doing it yet.
Domestically it’s so easy to take the train, you can cycle to the station and don’t need to disassemble/package your bike.
I think a taxi to the airport would be the only practical way to do this. My public transport trip to LHR is bus-tube-HEX (or bus-tube change at Green Park, which would be a pain carrying a bike) so not really feasible.