Say a Teddiursa is on an unstoppable combo of Fury Cutter. If someone uses a protect move, does Cutter's BP get reset?
Yes.
Power doubles with each successful hit, up to a maximum of 160 power. The power is reset if this move misses or another move is used.
Move flags: Makes contact. Can be selected by Metronome. Can be copied by Mirror Move. Blocked by protection moves. Power is multiplied by 1.5 when used by a Pokemon with the Sharpness Ability.
Source
Hope this helps!