I had my ILR or maybe FLR when our twins were born, but can't remember tbh. Was a hectic time: ) We were down at the London Embassy last year, around this time.
We took more documents down there than necessary, and everything was returned to us. I'd never heard of them keeping originals before. We actually made copies for them hoping it would take less time.
I took high school transcripts and had my military records scanned by my mom and sent over to us, just in case,
but they never even asked for them. They DID ask me a lot of questions like where I grew up and did I remember the highway that ran near where I lived. The gentleman interviewing me grew up in Ohio, and I'm from Indiana, after we started talking about areas we both knew well, he was satisfied.
They will not keep your originals, I also had to provide my prior divorce decree which to be honest would have also shown *proof* of my time in the US. Any also my older children's birth certificates, could have helped too, if they had wanted proof. (They were born in the US).
Best advice I can give you, is take extra diapers, toys, snacks, drinks for baby AND you. Cause you will be there a while. We were there for almost 5 hours. And the twins got bored quick.
It has been awhile and I'm going from memory, but all should be OK.
Oh and don't forget your little one, they will want to see her.
Have fun, I actually enjoy being around a room full of Americans again, until about 2 hours in and and the twins got bored! Then it was chaos! haha