Send someone to get help, even if it's hours away. The casualty's life may depend on it.
Someone needs to grab a phone and start walking. They should check periodically for signal, including at every high point.
I'd say it's unlikely they've passed out because of hypothermia, in the conditions you describe and consider the head injury to be serious. Intracranial Haematoma (bleeding on the brain) is life-threatening, not realistic to treat as a first aider, and has the very relevant symptom of progressive loss of consciousness. It's very compatible with loss of consciousness some time after a fall and blow to the head, such as after reaching shelter.
You still need to get/keep them warm, protect the airway (vomit is a real threat) and not make things worse. In the absence of any evidence of spinal injury, for the sake of maintaining the airway, put them in the recovery position with the head wound downwards (this may may help a little to relieve pressure build-up by allowing the wound to drain, but probably won't do much).
Another concern is blood loss. If they've been bleeding in the rain for some time they could have lost quite a lot. By the time they've reached shelter, they're unlikely to be in the process of bleeding out - either bleeding has slowed a lot or it's already too late. Again, if blood loss has led (or contributed) to loss of consciousness, you need professional help.
They could, of course, have been hypothermic for a long time if conditions were only a little worse than you say and they were hypothermic before they fell. In the Gansu Ultramarathon disaster earlier this year, runners set off with little or no warm gear climbing into bad weather. In a similar situation you may well have come across a casualty who has descended to warmer temperatures and reduced wind chill before succumbing to hypothermia and falling because of the resulting clumsiness. This means the hypothermia is more severe than I said above - but they still show signs of a head injury, and the care they need is still the same.
Thin synthetic exercise type clothes are far less of an issue than if they were caught out wearing lots of absorbent cotton. It may well be reasonable in the conditions you describe to leave the clothes. Once they're out of the weather, it's not that cold and they should rewarm easily. If we assume plentiful towels/blankets, you might wrap them in two layers, then try to get the wet inner layer out without uncovering too much. As you have access to water-heating, you can try to improvise hot- (or rather warm-) water bottles to provide gentle rewarming - ideally you'd warm the air but some heat tucked under a blanket near them will be of some use. Another person under the blankets with them is a good source of heat, even keeping a layer between you.
You imply that you've gone looking for someone missing in bad weather with "You have managed to locate them...". Doing this solo isn't a good idea, but you may not have much option. Doing this solo, without leaving word of what you're doing means you've neglected the first point in dealing with emergency situations: don't make the situation worse by putting yourself in danger. Of course there are limited situations in which this might not be the case, accidental separation from a hiking/running partner, but they don't fit your description very well. It's quite likely you should have alerted someone (mountain rescue via the police would be likely where I am) before going looking.
If it's just you and the casualty, I mainly offer sympathy - the decision of whether to leave them and get help, or stay with them and hope someone finds you soon is a very difficult one. Carrying a casualty is extremely strenuous, risky for both of you, and slow. It's not always impossible but is unlikely to be a good option.