In response to the question, you have become a British national after the birth of your child, so it cannot be simply passed on. British nationality is not retrospective; therefore only the child may be able to apply for indefinite leave to remain but this would be an issue depending on whether the mother is going to return to the UK. To be honest, it is better you get some specialist help.
Normally those children who wish to register must be free from immigration control and at the moment, I believe your child may not have an entitlement to register as a British national.