I flew into London Gatwick with my bike on US Airways. I read the fine print for their Transatlantic flights and found that if I simply taped my handle bars and pedals, I could take the bike along in place one of my two carry-on bags for no cost at all. They did try to get me to pay something at the airport, but I had a printed page with the airline policy and got my bike to Britain for free.
However, three months later I came back over with another bike, which I took on American Airlines (which we chose to fly for the PETS Program). American Airlines charged $100 to carry the bike over, and forced me to box it. The box cost $20.
I'd recommend US Airways for the bike, though check the website and see if the fine print hasn't been changed.