I’ve always wondered why do people say, “I’m going to hospital” or “I’ll be leaving for university in the fall” in Canada and Britain? Why don’t they use “the” before those two words?
It is just a stylistic preference. U.S. English has many such conventions as well. Your second example illustrates this.
In the U.S., we usually say “I am going to college” (which very well could be a college OR university). It is the same idea.
Generally it doesn’t much matter to the sense of “I’m going to (the) hospital” which hospital you mean, it is just the hospital-ness which matters. Adding “the” is just an arbitrary convention and different dialects will arbitrarily choose one or the other.
Britain: Where they have multiple maths but only one sport.
Going to hospital and going to the hospital mean two different things, likewise with University.
Going to University means you are a student. Going to the University means you’re a visitor. Going to hospital means you’re a patient, going to the hospital normally means you’re visiting the building for some other purpose.
And no-one in England is going to University in the fall 
And where sports teams apparently are a kind of They, as in Arsenal are pathetic this year and down in the “table” instead of standings in football, which clearly is not the sport Arsenal is (not are) playing. 
And they go on “holiday” while we Yanks go on “vacation.”
In the US we use the same distinction in regards to school. “I’m going to school” means I’m going there as a student to learn. “I’m going to the school” could mean you’re going to drop off your kid’s lunch, attend a PTA meeting or watch the football game, but you’re not a student. Similar for “college”.
We tend not to use “university” that way though. The name of the school might be “Such and Such University” but to almost all Americans in almost all other contexts, it’s “the college” or just “college”. Nobody says “I’m going to university” or “I’m going to the university” but they might say “I’m going to Southern Illinois University”.
Seems like a useful distinction and I don’t know why we haven’t adopted it for “hospital” yet.
I do it because it sounds more pretentious, as I am very pretentious.
IIRC, we’ve had this conversation before. Then, as now, a plethora of examples and counter-examples were given.
We’ve even had the discussion (also repeatedly, I think) about why Southern Californians refer to their numbered highways as THE 405, THE 210, etc., (e.g., “Take the 405 to the 5, then get off on the 210”), whereas in Northern California, numbered highways are usually referred to without the “the”. There’s no the there.
As stated before, we do exactly this sort of thing in American English (assuming that the perspective you’re asking from) except not with those words.
American English just doesn’t use the idiom of going “to university.” We say “to college” to indicate college or university. So, can you tell the difference between “I am going to college” vs “I am going to the college”? Or “I am going to school” vs “I am going to the school”? Or “I am going to church” vs “I am going to the church”? “I am going to prison “ vs “I am going to the prison.” It’s the same thing, only other forms of English also extend it to other locations like “hospital.” It distinguishes mere presence within a physical structure and being someone who is using the structure and institution it represents for its purpose.
Exactly this. In American English, “the” typically gets placed in front of nouns like this when we’re talking about going to / paying a visit to a particular location; when we’re talking about the more abstract idea of going to / attending a type of facility or institution, we use the “no the” construction (e.g., “going to college”).
“Going to the hospital,” in American English, is an exception to that rule, as we use the same phrase for both sorts of purposes.
Britain yes, Canada no. Everyone I know, save my Scottish mother in law, would say “going to the hospital”.
A similar use in the USA is, ‘I’m going to church.’ That phrase would indicate you’re engaged in a worship service.
‘I’m going to the church’ would mean you’re helping with a bake sale or food pantry or another activity that’s occurring in the church building.
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It doesn’t sound as pretentious when you use it with “crapper.” 
Here in Minnesota (and several other US states), nobody would have trouble understanding this conversation:
Adult: “where are you going to college?”
Young person: “I’m going to the University.”
Because they are godless heathens, of course.
Twenty years ago, I was associated with a local chain of test-taking schools in Bangkok. Sort of like Kaplan in the US. The emphasis was on students trying to get into graduate school in the US, so it was very American-oriented. I remember this one Irish staff member who was told in no uncertain terms that American style was the rule, but still he got his knickers in a twist over “the” before “hospital.”
Going to the hospital, means you’re going to a specific hospital. ie ‘I’m going to the hospital round the corner from work’.
Going to hospital is non-location specific. ‘My GP doesn’t offer that procedure - I’m going to need to go to hospital’
It’s really not that weird, compared to a lot of things we say.
In that example, the American English construction would be ‘My GP doesn’t offer that procedure - I’m going to need to go to a hospital’. 'The" in that case would most often indicate that the speaker had a particular hospital in mind.
Actually, in that example “a” or “the” would work – Americans routinely say “I need to go to the hospital” even when no particular hospital is meant – but you would have to use one or the other. That’s a British construction (deconstruction?), not using an article there.