You'd want to keep them awake for either or.
It's not for their health, though. It ultimately doesn't matter if they fall asleep. But treatment plans generally go a lot smoother if the patient is awake and talking. If a person has an allergy that only they know about, they'd want to know. Or if a person is HIV positive, healthcare workers would want to know.... And it'd be way easier to determine blood loss of the patient just told the healthcare workers. We're all trained to identify hidden injuries, but sometimes things aren't obvious for awhile. If a person was shot in the back and they're lying face-up, it might be hard to tell they were shot at all unless they were awake to say something. If you find a guy, knocked out in an alleyway, it might be hard to tell if he had a concussion or if he was just on drugs.
It's always much better if the patient is awake. Much much better.