For me I would say "to go to hospital" as the default phrase. As always, the definite article can be added if you have a specific one in mind - even if this is not explicitly stated. So -
Auntie Joan in Australia who we've never visited has gone to hospital with apendicitis. Here, in my usage, "the" could not be added.
"I've gone to the hospital with a broken nose" or "I've gone to hospital with a broken nose". For me both could be said but the nuance slightly different. In the first case we are talking about the one and only hospital I must have gone to because there is only one in the area and you know full well which I mean. In the second case this is just a general statement.
Edit - I also think I would say "to go to the hospital" if I wasn't entering as a patient. "I've gone to the hospital to pick up my aunt".
Edit edit - "to go into hospital" is also said, meaning that you are entering as a patient. "My aunt has gone into hospital".