United Kingdom - Passport & Nationality - Registration of Minor Child
The Registration of a minor child (under the age of 18) for British Nationality is possible if the child has spent at least 10 years in the UK. It does not matter if the child has been in the UK illegally during these 10 years.
If the parent of this child is also resident illegally in the UK, it is possible for the parent to make an application for a Derivative Resident Card, which will regularise their residency in the UK. The reason for this is that the parent's deportation would compromise the status of a child who has a valid claim to residence and nationality in the UK, and this would be contrary to their human rights.