Some more time hunting drafts. I managed to determine that the drafts are coming in from the adjacent space, so cut a long length of fiberglass to stuff into the space. And then went into the adjacent space to pull it into place, only to find a patch of it was cold. I mean, seriously cold, given it had only been in place for 10 minutes at that point. So I got down to look and found a little gap, maybe 1/4" wide and 2" high, with a seriously nasty draft blowing in. Cue the "you've got to be kidding me" response, and the application of Mr Hammer, Mr Chisel, and a load of cement to close up the gap.
Back in the closet there was still a slight draft that I traced, and the cement to seal that up is slowly hardening. It might take a little longer in these temperatures, but we'll get there.
At the hardware store I bought a length of fiber rope and a cigarette lighter. Lighting the end of the rope and letting it smolder helps find drafts - if the smoke does much other than rise straight up I know there's still a problem, and can move the end of the rope around to try and trace where the problem is. At present there's still something of a downdraft, but that's because I had to move a piece of fiberglass to do my cementing work and the draft is coming through the gap it left. Once the cement is dry I'll put the fiberglass back into place and hopefully that will seal everything up.
I also put tape around the windows in that other room. When we had windows replaced the guy who fitted them moved them from another room and assured me they would fit. And they do, kind of. They wobble side to side, to the point you can see an air gap between the window and the frame. I might as well have hired a monkey to fit my windows - the guy who did the work seems to have cut every corner he could possibly have cut, so I'm getting more and more unimpressed with him. Of course most people don't tear their house apart after having new windows fitted, so his average customer will probably never know what a hack job he does. Applying tape to the windows makes a remarkable difference. Now I just need to see how much difference - I'm going to run the heater and get the room up to temperature and see how well it holds.