When I had PHI with a previous employer we could only add dependents once a year. I would imagine you would certainly have to have a birth certificate to add a child to a policy.
I wouldn't worry too much, if there are, god forbid, any probloems which require immediate treatment, this would ALWAYS be done on the NHS. Private hospitals don't have emergency medical facilities, they have their place for non-urgent investigation and treating non-urgent problems where there might be a longer wait on the NHS, but for paediatrics/neonates, NHS is where it's at.
My DD needed emergency treatment after birth and the NHS were amazing, I could not have asked for better treatment.