When I gave birth, I just added on the birth certificate and they started paying me from there x
I forgot to update my uc about the birth of my baby untill she was 4 months (bad I know but they understood). Because I had child benefit set up already they will just check the details of Child benifit and when it started ect and use this as evidence of the baby and change of circumstances. We went in the other week due to rent rising and I asked if we need to do anything to prove change of circumstances due to baby and we don't need to do anything other than report it which I had at this point and they'll just work with CB, no appointment needed or birth certificate oddly.
After baby arrived and baby registered you can either go in with birth certificate or if you are in receipt of child benefits they can check your child identify from there
They won't actually do anything until after baby is born they'll likely ask you to come in for an appointment with the baby and their birth certificate before they add your child on. If they need you to do anything now they'll write it on your journal.