Private prescriptions are medication which your private doctor has recommended for you on a private prescription. A private prescription is not written on an official NHS prescription and so is not paid for by the NHS. The cost of a private prescription is met wholly by the patient and is dictated by the cost of the medicine plus the pharmacists charge for supplying it. A prescription is a legal document for which the doctor, who has issued and signed it, is responsible for. Therefore, a NHS doctor cannot convert a private prescription to an NHS prescription. A doctor you see privately can’t issue an NHS prescription.
A GP in the surgery at which you are registered can only provide a private prescription if the drug is not available on the NHS.