@Tibtom
Sounds like the private GenderGP was using the NHS as a free prescribing and testing service. Is this usual for private practice?
It's quite unusual, though it sometimes happens that after a private consultation a GP will receive a letter asking if they'd consider continuing a longterm treatment on the NHS. Sometimes this is completely appropriate, if it's a licensed treatment for a condition that has been properly diagnosed and is normally treated in general practice.
For something as specialised as puberty blockers or cross sex hormones, a GP should only be asked to prescribe in conjunction with a robust and detailed shared care agreement with the specialist involved. Such agreements are quite common in the NHS. Prescribing and administering GnRH analogues or testosterone to adults for their licensed indications is often subject to shared care agreements. It would be a brave GP who would take on the prescription of these drugs for the purpose of gender reassignment in children.