Around 3 months right now, maybe a bit more. The processing time is two months and then you have to factor in additional time for mailing. Normally the mailing is an additional 2-3 weeks. However it would be longer right now due to holiday mail volumes and also the number of holidays over the last few weeks where the post office wouldn't have been operating.
My family just went through this. They CAN get MSP even though the MSP website says you need the PR card for your primary identify document. I called them and they told us to submit the eCOPR, proof of address, and picture of passport. After those are submitted, you have to go to ICBC to have your picture taken for a BCID or driver's license. They won't give you the BCID or DL with your picture on it until your PR card arrives, though, but MSP will then be processed and you get an MSP card/provincial health number without your picture on it. Once your PR card arrives, you send a copy of it in and they will mail you the combined MSP card/BCID (or driver's license, if they have one).
