At 16 you can change your name without parental permission, so if a 16-18 year old wanted to change their name they could do that via deed poll and update their school records and the teacher would have no option but to use that name.
Obviously the parent would become aware of this due to school comms.
But there really are no circumstances where it’s appropriate for a teacher to keep a minor’s social transition a secret from whoever has parental responsibility (parent/guardian/social services) unless it’s part of a medical treatment which would then be supervised by a clinician (and a multidisciplinary team) and as the minor would’ve had to have several assessment appointments at one of only two locations (London or Leeds) the likelihood of that happening and the parents being oblivious is vanishingly unlikely.
IMO there needs to be some sort of audit for what and when parental supervision/sign off is required as it seems that while in some areas of life we have decided to slow down the child-adult process (eg staying in education/training until you are 18, paying child benefit until the end of key stage 5/age 19, expecting parents to contribute to university costs if household income is higher than x) yet in other areas adolescents are treated as mini adults (full price bus fare at 16, legal change of name without parents permission at 16, no need to inform parents if their young adult child drops out of the university course the parents are paying towards)
It makes absolutely zero sense that parents are required to give written permission before a school can take an under 18 year old to the nearby art gallery or for the school to snap an under 18 year old’s photo for the school newsletter, but the same school believes it can facilitate a change of an under 18 year old’s sex marker on their school record without even telling the parents about it.