I expect it will be on the registrars & NHS databases rather than needing to actually see the bit of paper - I remember needing evidence that my daughter was registered at the local GP for infant school admissions* (and the NHS number issued at birth and has an unchangeable sex marker - adults who transition have to get a whole new NHS number to have a new sex marker - which is another can of worms but I think we can safely assume that no one in the UK has transed their child’s paperwork before the school admissions process begins at 3 and a half. At least not yet)
the only way I can see a single sex state school for kids admitting an opposite sex child is if the child was born elsewhere and has migrated here (some US states allow a child’s birth certificate to be changed and the US government are allowing x on passports so presumably they don’t care about biology) or an in-year transfer where the application is made directly to the school and the leadership team are all very pro trans. That would be a massive risk if complaints are made in future/anything goes wrong for the child or their classmates.
Private schools are going to be more variable but (at least in the smaller ones) they tend to be conscious about the bottom line and getting the fees in, so if admitting one opposite sex child risks losing more than one existing pupil from the roll then they are unlikely to consider it. Private school parents are customers after all, and there aren’t that many potential customers in the pool to risk pissing them off.
A school admitting a male with a gender identity to a girl’s school will be open to legal challenges (technically boys can attend a girls’ school in really limited circumstances but it’s a carve out designed for scenarios like ‘the head teacher of a remote mountain top boarding school for girls has a son of school age and there is nowhere nearby for him to enrol’ and even then he’d be expected to share sleeping/toileting facilities with his teacher parent and wouldn’t just be a standard student alongside all the other girls outside of the classroom)
*I’m applying for a local state girls high for my daughter through the usual transition from primary to secondary admissions process. Deadline is in a couple of weeks (still deciding on 2nd/3rd choices) but I will report back once I have submitted.
Where I live the admissions process is so cut throat that schools criteria/proof of residency has developed in response to parents trying to game the system (lots of private prep schools and tutoring for the 7 grammars and then sharp elbows for the high schools in the posher areas). There is no way parents round here will tolerate a male child taking a girls place at a school or vice versa, so it’ll be interesting to observe how this plays out.
As an illustration here is the residency info for one of the two girls’ grammars (very prestigious, very genderwoowoo, teeming with transboys, so female children. And rightly so, because once a child has been admitted to a single sex school for their physical sex it would be against the law to ask them to leave due to their gender identity/gender expression)
(obvs I won’t be sending my daughter to the genderwoo girls’ school! Her preferred school is very ‘Daughters of The (name) School Family’ and ‘Seven Years of Sisterhood’ which seems quite old fashioned but also is unapologetically female-focused, so I can forgive the fustiness!)