It sounds so precious to hear Britishers talking about going to hospital. Here in the USA we would say we're going "to the hospital."
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But that would entirely depend.....
If I were talking about someone who was ill, I'd say that they had to go to hospital to get it sorted out.
But if I were talking to one of my medic friends about what time their shift started or asking what time someone's appointment was, I'd ask them what time they had to be at *the* hospital.

I'd guess that in the first case that would be because I wasn't necessarily sure which hospital was being talked about. Whereas in the second I'd be referring to a very specific hospital.