Thank you for your question
My name is Clare
I shall do my best to help you but I need some further information first
For clarity - is the mother (or father) British by birth or by descent?
In that case the child is entitled ti a British Passport and to live in the UK.
All the mother needs to do is apply for a passport for the child
There is no need to Register the child in the UK
Please ask if you need further details
That is hard to predict.
It would make it easier to get a copy of the birth certificate if it was ever lost, and of course would mean that they can always prove their nationality even if the passport is lost