After you finish BMQ there are several paths:
A) If you're combat arms. you'll get transferred to the PAT platoon of whatever Battle School your trade uses (ie. Combat Engineers get sent to the Engineer school PAT platoon at Gagetown, RCR-bound infanteers to the Land Force Central Area Training Centre (LFCATC) PAT platoon at Meaford, etc etc.). The time you wait on said PAT platoons also depends on how often the SQ and DP1 courses are being run, it can vary, ie: buddy of mine in meaford started his infantry SQ/DP1 within days of arriving at meaford, another is a combat engineer who had to wait several weeks before starting his SQ in Gagetown.
B) If you're an army support trade, you get transferred to whats called Post Recruit Education and Training Centre (PRETC) in Borden ON. There you wait until you are put on the next available SQ course, or possibly other pre-requisite courses (such as drivers) if you so require them.
I was put on PRETC for a week, than I did my Driver Wheeled course, than the day I finished the driver course I was sent off to Meaford for SQ. Some people had even shorter wait times, and were sent off to SQ within days of arriving at PRETC. Others were....not so lucky and had to wait a few more weeks.
And sometimes there isn't enough time to put you on an SQ course and you just get sent straight to your 3s. Rare but it happens.
So really it all depends on how many courses are being run at the respective training centres, it's difficult to say exactly, but based on what I've seen at PRETC, you're looking at a 2-4 week wait on average.