United Kingdom - Permanent Residency - Derivative Residence Card
If a child under the age of 18 has been resident in the UK for at least 10 years (even if this residency has been illegal and without a valid visa), it is possible for the parent to make an application for a Derivative Resident Card. This 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.
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.