The sources I have checked contain no dire warnings about falling asleep in cold conditions; my impression is that it is an old wives tale. If you are hypothermic and your condition is not corrected as your body temperature declines, you will lapse into unconsciousness, leading to death.

My own experience when I have tried to sleep with inadequate insulation for conditions, is that I will fall into a fitful sleep and then wake from the cold, sometimes shivering. On the worse such occasion, I melted snow, made a hot drink, and dropped off to sleep again, repeating the cycle through the night. I had notably more energy the next morning, despite sleeping very little.

The main thing is to recognize incipient hypothermia, take measures, and let sleep take care of itself.