I flew back direct from Phoenix Skyharbor, Arizona to London Heathrow with BA, and they were absolutely terrific. Going through customs Stateside, I attracted only amused glances and a few questions from staff, including an "Oh, I love Paul McCartney" ! When I got on the place, the staff ushered me to First Class (I was obviously travelling cattle class!) and put the bass (in a hard case) in the coat cupboard. 9 hours later as I exited it was waiting for me by the door. Fantastic, easy service from BA, despite the idiots on the phone not having a clue what to suggest to me before I went to the airport.
Good luck!