This may be of interest:
viewtopic.php?p=1705061#p1705061
I arrived at Heathrow before mid-night in June and rode into town on the A4, which has cycle lanes off both east / west carriage ways. Not exactly pretty, but acceptable. At mid-night, I comfortably rode the A4 with no hassle at all (to Brentwood, less than half an hour); you would just keep going via Hammersmith, Kensington etc. All fine at that time, and do-able in daytime too.
I always use the Tube to Heathrow and never bother with the rip-off Heathrow Express, unless someone else is paying (or the even more egregious Gatwick Express: every Brighton train stops at Gatwick; it is utterly pointless, and just catches out foreign visitors).
When you say 'unpack' your bike at Heathrow, do you mean remove the packaging and assemble it? That makes it less attractive to taxi drivers. In April I brought a bike back from Heathrow in a mini cab (just a Mercedes Saloon in fact), and the cardboard Halfords box fit in the back not problem.
(You won't be able to get your bike on a Tube at Heathrow; I was stopped at Hatton Cross and told - utterly implausibly - that if I got on at Hounslow West, which is an above-ground station, I would probably be allowed. Get away with it seemed more likely. Of course, just carrying a box, you are fine on the underground.)
London traffic is far safer for cyclists in my view than the countryside, and I ride in Hampshire at weekends (ie today). There, a gormless car culture still prevails and cars drive aggressively fast and close to cyclists. Not in London: drivers know the cyclists will catch up at the next junction, and every car number plate is clocked.