Agreed with the others that you are going to be pushing it to get the birth registered, go to the embassy to register the birth and get US passport and get the UK passport. You'll have to do all of those--you know you will need both passports, right? It's a lot to do when you will be tired after just having a new baby, but if you want to push to get it done, more power to you.
All that aside, I booked an international flight while I was still expecting my little one. It was just an EasyJet flight, but I'll share my experience. I decided to telephone and ask what to do before I booked any tickets online because I was worried about the fact that we didn't have a name yet, and we could always be surprised to have a boy when we thought it would be a girl. They advised me that I would need a name, and would have to pay to change the name if I booked the tickets and we named the baby something different than was on the tickets. I believe they had me do my tickets online for the rest of the family, but I was able to call in and add her to my ticket later without incurring any additional charges. I'd check with your specific airline first though to be certain.
Do also call once you have the tickets and try to reserve the bassinet seats. You then get the galley, so you have more room for baby stuff, and that cot is a life-saver!! Sometimes you can reserve it in advance, and sometimes it goes to whoever reserves it first on the day, so you'll want to show up early at the airport.