If you're asking what would happen in the real world, in an experiment, then nobody knows, because it is not possible to make the whole universe rotate around the bucket in a way that is different from just putting the bucket in rotation w.r.t. Earth and then changing the frame of reference to that of the bucket.

In other words, Mach's idea, while interesting theoretically, and stimulating thinking and development of theories, is extremely hard to study and check or disprove experimentally. It's on the borderline of unfalsifiable.