I'm undecided whether to put my ex on the birth certificate, but either way he won't be able to go until a later date anyway because he'll be away for the first 4 months of their life.
As for the CSA it can take up to 26 weeks to sort out and I don't think you even need to wait until the baby is born actually. I could be wrong, but it's worth looking into. I need to do it myself actually.
This might help https://www.csa.gov.uk/