Is there a general convention as in the US (i.e. "until April 1"), or is this highly institution-specific?
Its completely context specific. It might even depend on the quality of the applicants - if I have two good candidates, I might want an answer from the first choice quickly so as not to risk the second choice taking up another offer if the first choice says no.
Conversely if the first choice is really good and I really want them, then I am more likely to be willing to agree to wait for an answer, so as not to risk them being rushed into saying no.