So, just an update. I ended up taking my iCandy pram with the car seat attached to the airport. If they were going to make me check it at the desk, I was going to leave it with DH and just take the Baby Bjorn. But, they were super nice about it. Had no problems letting me gate-check it. Even going through security, they didn't make me take DD out. When I got up to the gate, they let me bring my car seat onboard and put it in overhead luggage. The stroller was not waiting for me when I landed (which kind of sucked, because I had to haul everything through customs and immigration), and I was worried it was going to get broken on the carousels. But, then I noticed all the strollers from our flight had been brought out and were sitting between the luggage carousels waiting with an attendant.
It was fine, and there was someone else's iCandy waiting there, too. Nothing was broken and it couldn't have been easier. I'm sure it all depends on the airline, but in this case, BA was great.