There is no substantial difference between applications for children and applications for adults, except that for children, there will definitely be no NINO records that prove their UK residence. That being said, the same principle applies:
1) Use their Passport to prove identity
2) Refer to
this list to select the most appropriate documents to prove residence (very likely the best pick are their school records, one per year).
Is there a specific aspect of the application that you are unsure about? Otherwise, all info
here.
My posts express what I believe are the facts, based on the best of my knowledge, about the topics discussed in this forum. They do not constitute immigration advice.