After you have the baby, you'll call your local registry office to set up an appointment. I could go to a couple of different ones. I called around and found out that if I registered at my local council, I'd have to wait for the birth certificate to be mailed to me (10 days I think?) or I could go to the main one (larger city) and get the birth certificate on the day.
It will all depend on availability around when you deliver. You have 42 days to register the birth. Some may have quick availability, some may have a one month wait. It's hard to say!
Honestly, it'll be a tight time frame. You might be fine for travel at Christmas but you might not. You could deliver early or you could deliver a couple of weeks late. You have a child, you know how unpredictable these things are!
You'll have so much red tape and beauracracy to go through afterwards, it may just save you a lot of time and stress to plan a trip to the US a few months later...