I'm sorry, I misread your initial post. It is correct though that MN1 is not a visa application.
Here is the correct guidance for this case
https://www.gov.uk/government/publicati ... plications
Expectations for a child who has lived in the UK for 10 years
Compliance with immigration law – child who has lived in the UK for 10 years
We would normally expect the child to be lawfully in the UK. To grant citizenship to a child who is here unlawfully could potentially undermine the immigration system, as it could be viewed as rewarding or incentivising non-compliance. However, we recognise that there may be cases where the unlawfulness was beyond the child’s control.
If the child is not here lawfully you must consider whether there are exceptional reasons to grant, in which the reasons put forward for granting citizenship outweigh the need to promote compliance with the immigration law. In considering this, you must take into account the age of the child, the connections they have established with the UK, their length of residence, and their particular circumstances. (For example, it may be appropriate to register an older minor who has lived in the UK since they were a baby, has completed all of their schooling in the UK and has demonstrated very strong personal connections with the UK through relationships and involvement in community groups.)
For younger children (who are not in local authority care) who are in the UK unlawfully, you must not normally grant the application, unless there are exceptionally compelling grounds that justify moving the child from being here unlawfully to becoming a British citizen. Such younger children are not usually at a critical point in their lives where they might lose out on opportunities.
Parents’ status - child who has lived in the UK for 10 years
We would normally expect the child’s parents to be here lawfully, as this means that the family’s future is likely to be in the UK.
It may be argued that for a child who has lived most of their life in the UK and formed connections here, it is in their best interests to secure their status. However, this is more apparent for older minors who have formed their own independent connections (which they would expect to continue were the parents to leave the UK). A younger child’s future, however, will normally follow that of the parents, and so their future intentions may not clearly lie in the UK. The best interests of a younger child could be to follow their parents’ residence and status.
In the case of an older teenager, you must therefore consider whether the arguments put forward about the child’s interests and strength of connection outweigh the fact that their parents are not here lawfully. If so, and the other expectations are met, you may grant the application.
For younger children, if the parent’s status in the UK is precarious (for example, if they are here unlawfully, or are subject to removal action), the child’s future may not lie in the UK. You must weigh up whether the child’s circumstances are such that there are significant grounds to register, which outweigh the fact that their future may not be in the UK.
It is not clear to me how would having overstayed a visa be considered for a child in these circumstances. As child is over 10 years old they will be expected to respect the good character requirement - going off that, there must be no other factor weighing against the child's character for this requirement to be satisfied. How long was child's overstay, why did the parents not secure a visa for this child?
All advice comes from personal research and experience and should not be regarded as professional opinion.
