You'll get promoted right after BMQ, probably on your grad parade actually. You'll get back pay to make up for the time spent as a Pte.
BMQL has nothing to do with getting promoted to Cpl, you'll be a Cpl on BMQL one way or another.
Think of it this way: You need BMQ to do QL3, and need QL3 to get your badge. That's it. BMQL has nothing to do with becoming an MP, it's simply a career course you'll have to complete sooner or later to be promoted to MCpl. Getting it done earlier would be better, so you don't have to worry about it later on down the road.
It can be confusing because other trades you need BMQL to be promoted, but BMQ = Cpl. QL3 = Cpl with badge, start doing patrols at a Det. BMQL is simply a career course, like QL5, QL6B, etc. 0 relevancy with becoming a Cpl or getting your badge.
You could be a Cpl with your QL5 and have 5 years in the branch, and would not hear about BMQL until they want to promote you to MCpl. And then comes your PLQ, but that's a different story.
(the only reason we do BMQL is because we are a 'purple' trade, as in you can be Air Force, Army, or Navy in the trade. Our trade doesn't belong to any single entity like the Infantry does to Army, Pilot to Air Force, etc. Same as Supply Techs and the like. BMQL has nothing to do with being an MP, simply a career progression course to move up in rank)
Hope that makes sense. Feel free to PM me any questions.