That'd depend on the hospital building out an RS485 infrastructure to support the device, though, which is a nontrivial ask.

Piggybacking off of a cell network is honestly probably the right solution. It doesn't have quite as high resolution as GPS, but that's hardly necessary for the sort of timestamps we're discussing -- even a resolution of a minute or two would be a huge improvement on "hope the tech who installed the equipment set the time correctly".

Remember that anything that is harder than setting the date and time is unlikely to be done in practice, since we're looking for something easier than that. Routing cabling out of a room seems unlikely.