They include care home worker Ray, who weighed 148kg or 23 stone when he began taking Wegovy in July 2024. He has struggled with his weight all his life.
Ray needs two operations, but doctors have said he needs to lose weight first.
With so many patients meeting the criteria, the hospital is prioritising those like Ray who need surgery or who have multiple weight-related health complications.
Here, not only is Ray given the drug, which is taken via weekly injection under the skin, but he gets face-to-face support from doctors and dietitians - advice not always given to those buying the drug privately online.
They stress the jabs do not do all the work and it is important that patients change their lifestyle, and eat healthier food and smaller portions.
Ray is joined at the appointment by one of his daughters, Sophie, who says it would be "amazing" if he could reach his goal of losing three stone: "I wouldn't recognise him. It would be like I have a brand new dad."
For now, the drug is available on the NHS in England via these specialist services, mostly in hospitals.
But the chances of getting the drug are low.