Would getting a faster MSc instead of just BSc affect admission chances for PhD?
(I'm on the admissions committee of a top-10 US computer science department; my advice may be way off base for mathematics PhD programs.)

All else being equal, getting a master's degree in one year would probably count against you.  You're far better off taking some graduate-level courses, and possibly getting some research experience, while staying in the BSc progam.

At least in my department, PhD applicants who already have master's degrees are held to a significantly higher standard than applicants who only have bachelor's degrees.  While having formal research experience is an advantage for undergraduate applicants, it's a de facto requirement for applicants with an master's degree.  One year is not a lot of time to get some real (meaning publishable) research experience, especially since most one-year master's programs have heavy coursework requirements.  And you'll be competing with other MS applicants who've already spent two years in graduate school.

My department steers PhD-oriented undergraduates away from our 5-year BS/MS program for this precise reason.