Can we say one of them is more American English or British English?
I have to go to school.
I have got to go to school.
I got to go school.
I gotta go.
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From an Australian English perspective (which is closer to UK English)
- I have to go to school.
Very common and makes sense.
- I have got to go to school.
Could be used to emphasize that you need to go - perhaps because you are worried about being late?
- I got to go school.
Grammatically incorrect. You need "to" in there: I got to go to school. That is slangy, but would be a common way to say it.
- I gotta go.
Very casual (and doesn't say where you are going)
- Gotta go
More of a farewell. Could be rude?
From an American (somewhat Texan) English perspective:
I can't really say if there are any big differences between this and British English. Comparing with Brendan's Australian English perspective, I'd say that there aren't many differences. Or at least that the differences are trivial.