Welcome to the board!
Unfortunately, from what I read the British airlines are really strict on their policies and very against car seats.
If you read it on their website, that's probably what will happen. (One airline, can't remember if it's BA or Virgin, even requires you use THEIR child seat instead.)
I can think of a few possibilities for you. One is the CARES harness. While it isn't as ideal as the car seat, since sometimes the lap belt doesn't tighten well and a child can still unbuckle it, it may be better than nothing, and my daughter slept ok in it once her seat was reclined. It is approved in Europe, and since it's not a traditional car seat, they may be ok with you using it. It's worth calling British Airways to see what their policy is on its use. Here's the website:
http://www.kidsflysafe.com/ They have a page on using it in Europe, and it says the UK Civil Authority approves it, so you can use it on any British airline. You would still need to make arrangements for a seat on the other end, since it's not safe for your car seat to check it or gate-check it (it could be lost or damaged.)
If your tickets are refundable, another option is to cancel them, get a refund, and buy tickets on a US-based airline instead, since US-based airlines are required by FAA regulations to let you use your car seat. Of course this may not be possible or practical in your situation, I don't know, but I thought I'd throw it out there, as if it IS possible, it would both help her be comfortable, and keep your car seat safe and with you.
Good luck in whatever you choose to do! Please let us know if you have any other questions, too.