The diesel supply in all countries is dependent on the grid, so days is the absolute maximum. The reality is often much less. During the recent power outage in Portugal, the Alfredo da Costa maternity hospital had only one hour's diesel and had to be resupplied by ministerial chauffeurs delivering Jerry cans of fuel.

Still, all the ancillary services that go into a hospital like water, sewage, medical gasses, cold chains, etc are all dependent on the grid, as are the people who make them work. If a large-scale outage happens, most hospitals will start losing patients within 24-48 hours and will close as functioning hospitals well within a week.