I can think of a couple of tricks that might work:

First, the mummy bag can roll with you; try rolling it over so the hood is around your head, only "upside down" to a back-sleeper. (I sleep on my side, so I roll the bag with me so the hood opening is "sideways.") This doesn't solve the problem of what to do with your arms, though.

You might try sleeping in a hooded down jacket (Patagonia and Montbell, at least, make them) and a hoodless sleeping bag (I like the WM Mitylite.) The hood will then move with your body, while the bag stays in position. This comes very close to sleeping under a quilt (as Tango61 suggests), but the hooded jacket provides some warmth for your head.

Depending on how cold "cold" is, you might even investigate an "elephant's foot" bag that would cover you from roughly the waist down, coupled with a heavy hooded down jacket. I'm only familiar with the concept, but don't know who (if anyone) still makes them, or where you could have one custom made.