OK, I decided to give this a go on my lunch break. The kitchen was running about 60F. I forgot to check what temperature the water came out of the tap at.
I had the equipment:
I got it lit just fine:
...and then I waited, and waited, and waited.
Finally after 55 min and the water only reaching 136F according to my meat thermometer, I was out of time, so I dumped the water into my pot of chopped potatoes and used my burner to get the water boiling and the spuds cooked, in barely 15 more minutes.
So, if you have no other option, this will give you a warm drink using a bullion cube, or some homemade broth, or some home dried herbs.
It would also give you warm enough water to fill a hot water bottle to take the chill off the bed.
It will *not* be a practical way to avoid a boil water advisory - just too slow.
It will *not* be a practical way to actually cook anything.
You would be so much better off to make sure you had some dry firewood stashed and build yourself some sort of mini-rocket stove out of bricks. Yes, this would have to be outside, and those little stoves have to be constantly fed, but at least you'll get some serious heat out of it.
So this does *not* pass the permie test overall, but if it's your only choice for a warm drink, it would be better than nothing.