My understanding is that with severely hypothermic patients you want to keep them awake if possible. I'd heard moving a hypothermic patient is risky, as you don't want to jostle them. Some wilderness survival experts say to stay put if you're lost since SAR is likely looking for you. How important is the "not sleeping" advice and why?
I am not a doctor, so I can only repeat what I think I understood from lectures by those that do have medical training.
I'm pretty sure I remember Dr Murray Hamlett (I highly recommend attending one of his lectures, if he's still doing them. He is not only a leader and pioneer in cold weather medicine, but also a very good and engaging speaker.) saying to keep the patient awake. I do not remember exactly the reasons he might have given, but keeping the internal metabolic activity as high as possible sounds like an advantage to me. Basically, we burn fewer calories, and therefore produce less heat, when asleep. It is also more difficult to assess a patient's state when they aren't fully conscious.
Never allow them to sleep. Even if hypothermia is not that severe. I'm talking from a personal experience where two fellow trekkers died due to this. The reason is when you go the sleep, your breathing slows down due to low heart rate. Our body is meant to behave so. This means the blood flow slows down. And with a mild/severe hypothermia, you are in a state where the vital organs might not end up with the required blood flow. Keeping that person warm and trying to improve the metabolic rate is the best solution when you don't have medical help at hand.
I would take two things into consideration, or rather mention two things that are in consideration:
Providing a pleasant, agreeable, and accommodating-as-possible response to someone who is suffering.
Doing what will provide the best chances of saving a person's life.
American culture at least is squeamish about violating principle #1, enough so that someone who violates #1 to care for #2 will feel very unpleasant and perhaps guilty. But #2 should trump #1 if you have a choice between the two. It runs on the same principle as an intervention for an alcoholic: it is not in any remote sense pleasant for the person, and done right it is one of the most painful things you can experience (and the other people in the intervention experience, and hurt for, the person receiving the intervention), but it may be the most loving possible thing you can do.
I do not have credentials to offer medical advice for how safe it is to allow a hypothermic person to sleep. I personally would find it very unpleasant to force contrived activity to someone who is entirely drained and just wants to sleep, and may sleep forever. But if the best medical chances for survival are for preventing sleep, the #2 listed concern trumps the #1 concern, period. Feel guilty about it all you want, feel like you're torturing a poor, helpless bunny, but pursue the hypothermic person's medical best interest as far as you can.
If he has passed through the shivering stage, you are probably going to lose him unless you can get an external source of heat.
If he's at the 'umbles' stage then food, hot drinks, and keep moving it possible is a better course of action.
If you have him warm enough that he has become lucid, then let him sleep.
Hypothermia beyond moderate cases (uncontrolable shivering has ceased) is hard to deal with in a remote situation. Prevention is far easier.