How old is the baby? I assume you'll be alone?

I was able to carry mine in a good quality baby carrier up to age 3 and mine were big for their ages. I did that in the airport, put the car seat on a metal luggage cart and pulled it behind me. The only carry-on I had was a backpack. At age 4, they had their own backpacks with stuff. An extra bag can be hung off of the car seat if necessary (for me, only when I had more than one in diapers).

So no stroller to fuss with and I could get on and off the aircraft faster with the baby attached to me. I simply held the top of the car seat with the cart still attached to get down stairs.

In the terminal, a toddler could ride in the car seat (from about 18 months on).

BA and AA are code shared so that should make it easier.