Many people assume that a gp has to prescribe meds suggested by a consultant (NHS or private), but they are not employees of said consultant, and are legally liable for any medicine they prescribe.
My surgery had a policy that you had to have at least one private prescription, to check everything was okay, before an nhs prescription could be written. Also, have you checked the price of the medicine? It may not be as bad as you think.
Also, areas have formularies and certain drugs can’t be prescribed by gps, and have to be prescribed by consultants. I’ve just checked one formulary and it does say ‘specialist initiation’, so it’s whether the gp accepts a private referral letter.
Finally, it may just be a time factor. Did the consultant email or post the letter? Then the letter has to be scanned on the system, read (by admin staff usually), patients records updated, then forwarded to gp to read and approve and issue. This can take time. A normal repeat prescription can have a 48- 72 hour turnaround.