When I was a houseman my legs were blebbed and bubbled by the bites of the fleas from the feral cats that lived within the hospital grounds — but things have moved on.
Hospital cats are now more professional — they have come in from the cold, are properly trained and equipped, have regular health checks and probably mandatory immunisations (long before the other staff!)
The thing about cats is that you cannot exclude them if they don’t want to be excluded and some cats just have a vocation! So the NHS works with them, not against them.
Somewhere in Britain, probably everywhere in Britain, Buster here (or a cat like him) guards the confidential waste in the office of the emergency department. He keeps his wits about him as he dissipates the stress of staff, his mouth shut, the perfect confidential mentor and counsellor but as a member of the occupational health psychiatric team, as you can see, he does have his own panic button.