In a hospital setting nobody will be degloving your finger - this is a common situation, and they have pliers that will crack/shatter the tungsten for removal.

I'm guessing that strongly depends on how big the hospital is. I know someone who lives out in the boonies and it's amazing what isn't on hand.

I live in a town of <15k people, and it’s the county seat. Our hospital can handle tungsten carbide rings without issue. I checked before getting one for this very reason (though I now wear an Oura as my wedding ring).