HI, you almost, but not quite got the idea of a cagoule.
If you have ever walked through the brush when it has been drizzling for a couple of days you will understand.
Old cagoules were rain gear.
Like Haley 1 says above, a poncho.
Except it is sewn together like a pullover.

They were made big enough to drape over you even with a pack on.
They were also made long enough to reach your ankles like a dress.
The reason they were so long was that every leaf on the shrubs holds a drop of water and it would soak you from the waist down as you walk through it.
Even if it was not raining enough to get your head wet your legs would be just soaked from the water on the low shrubs.
Today we would wear rain pants or leggings.
The cagoule was cheaper to make and better because the water would not run down your legs and into your boots as badly.

The time when you would pull the cagoule up around your waist would be if you were walking down a road or someplace without any bushes to brush against you, just to give your legs more freedom of movement.

The old cagoules I saw were oilcloth.
That is a light canvas which has been treated with linseed oil, a lot like those Australian Outback coats.
As for lighting a fire under a cagoule to stay warm, umm no. It would not work.

However seeing as a cagoule was rain gear and not a parka that means it is not bitterly cold and if you are wearing wool shirts, pants, and a sweater you shouldn't freeze and might even stay fairly warm. It would be almost the same as sleeping under a canvas tarp.
You could stuff it with anything bulky and dry too. (if you could find anything dry)
Not at all comfortable, but doable.

Yeah, like Haley says, it is the same as sleeping in a poncho.
I would not do that if I had a tarp or a tent.

The tent thing that Desperado showed has a configuration a lot like a cagoule when it is not pretending to be a tent.

Edit:
I do not own a cagoule. Whenever I have to travel through soggy brush like that I either wear rain pants or I make trash bag leggings.
It might look funny but if it keeps me 100% waterproof and warm I am all for it.