Re: bound to be poor data
"If a school asks for nationality data, do they then ask for proof?"
Yes.
"Since we don't have ID cards or compulsory passports"
Doesn't matter. Like all the hostile environment policies, you have to prove who you are in order to be entitled to have it.
"Why can't I use NHS services or state schools telling them I'm John Smith?"
Because fraud.
In the same way I can't go to the bank, claim I'm Boris Johnson and take out a loan for a couple of million for pocket change and pretty hair.
You can use emergency services without giving ID (I hope), but anything more than that it's ID time.