Asking University for quick admissions decison So I have received an offer from a school (University A) which requires I send my decision to join them by a deadline, coming soon. While the other school (University B) still hasn't decided.

Should I inform the admissions committee of University B by email so that they can send their decisions quickly? If so is it a better idea to share with them the admission letter of University A? University B requires a supervisor that approves students for admission. Should I tell my prospective supervisor about this?
Asking for a definite quick decision might, in some cases, result in a rejection. This would be the case if the university has strict rules about admissions and about treating all applicants fairly. There is no guarantee of that, of course, but it seems more likely than a quick decision to admit. But different places have different rules and procedures. In the US, for example, every university is unique. 

But you can certainly ask for the status of your admission and what is the earliest you can know. You can say that you have other decisions pending, but I wouldn't be precise about them. On the other hand, I think you can be more forthcoming with a prospective advisor. They might be able to at least get access to information that you can use to make a decision. 

You can also ask for a small delay from the place that has accepted you, saying that you need some additional time to consider the offer. The likelihood of refusal of the request is probably high, but not absolute.