Thank you for your question
My name is Clare
I shall do my best to help you but I need some further information first
Has the baby been born?
Then that is fine - you can Register the baby in this way if you wish.
Please ask if you need further details
It is matter of who registers the child - if you do it first then it is your choice that stands
Neither of you has more right and the Registrar is likely to suggest that you go away and decide what you want between you