Not the City but the name and address of the hospital (assuming, of course, you were born in hospital).
I ask because I am filling in a form which asks just this. Now I was born nearly 60 years ago in a non-UK but European city (I am British). I do know the hospital was knocked down about 50 years ago but as Mum is no longer with us and my Dad is very confused, I don't have a clue which one. Google tells me that there was a mass building post-war clearance in the 1970s but, oddly enough, not a list of all the hospitals (sarcasm). I was registered at the British Consulate which just lists the city name as place of birth.
The form which asks for it, is an important document I need for a new job (bit like a dbs) but apparently this piece of information is vital! The office I spoke to is checking with a higher authority but in case I need to argue my case, I just wondered if this is a piece of information that everybody else is aware of.