Since before my time it varied province to province as to what was allowed.
From 87-2004 you where still required to take Driver Wheeled before operating any CF vehicle - Civy license or not - however some provinces you where not allowed to drive on a road in a CF vehicle (base use only) if you didn't have a civilian license.
Oddly I taught a FMC Driver Wheeled (back when FMC was a thing) and some of the other Driver Instructors didn't have civilian licenses - but where allowed to off base instruct but not drive (things that made you go hmm).
There may be differences by province, because that is not how it is done in Ontario. A road test is required for each different category of civi pattern.
Some units tended to fudge that - but it's on the Driver Examiner and (if different) Transport NCO if that backfires.
We had a guy back into a telephone pole in a 15 pax van in Petawawa getting stuff for the unit canteen - and it was unpleasant for many when it popped up that he was never tested on that.
Safe Backing used to be a yearly refresher - non vehicle specific, same with Dangerous Goods.