The govenment owns it, but under the PFI agreements introduced by John Major, and the used profusely by Gordon Brown, the PFI companies built the hospitals, and rents them to the local health authroties on a 25 year payment lease, at which point the buildings themselves are then the ultimate property of the health authority in their entirety.
The NHS occupy and use the buildings, but are paying a much higher rent than if they'd built the hospital themselves. Brown did it to fiddle the books, and keep the cost of them "off" of the balance sheet, so government borrowing didn't have to rise to pay for them.
You wouldn't be renationalsing the NHS, just certain buildings.