I feel this one's fairly arbitrary, but "taken to hospital" at least lines up with how we talk about other public places in a generic, consumer-of-public-services sense in British English. Besides "taken to hospital", you can be "at university", "on your way to school", or "in prison", for example.

If you worked at one of those institutions, though, you'd add an article and be specific - you work "at the university" or "at the hospital". (You'd expect the listener to infer that you mean the specific university/hospital in the area, assuming there is only one.)

Edit: More colloquially, we also have "going into town" (no article, even though it's probably a specific town that is clear from context!)

The article-less usage only seems to work for generic places (hospital/school/prison/university). It doesn't belong with a noun that refers to a person but also sort-of-refers to that person's place of business. You're "going to the doctor" or "the dentist" or "the hairdresser" - never "going to doctor". This is despite the specificity often being superfluous here - the listener might not know/care which dentist you use!