But would expand on it somewhat.
The 'proper' comb change methods (search for "shook swarm" and "bailey comb change") do depend on the existing box NOT being a mess of wild comb and broken, weak or non-existant frames.
If you cannot safely (to the Q in there) remove frames in the "minging" box, I'd suggest the following refinement to what you suggest.
Put another (but decent) shallow on top, get them to move up into that, and when you see Q in the new box, stick a QX between the boxes (trapping Q in the top box). Then 3 weeks later (after all the brood in the rotten box has emerged), you can scrap that bottom box. And anytime after you have Q on a decent frame in the decent box, you can start the process of moving to full-sized brood frames.
Using a shallow should be much quicker and easier for the bees to draw out than a deeper box, and Q will therefore go up and start laying in the nice box very much quicker.
Make it an insulated (or poly) box on top to speed the work.
Make it a used (fumigated and aired) box to be more attractive (and quickly accepted) to the bees.
Sacrifice some drawn frames into that box to make it really quick. (More people have available drawn shallows than drawn brood comb!)
I'm going to be prescriptive and say that the new box should have hoffman-spaced frames on rails. Its going to be a brood box for a while, and you are going to be inspecting it to find Q when she is upstairs. If you don't have hoffman shallows, invest a fiver in some hoffman converters - not ideal but good enough for this.
Wait a while (a month? in Edinburgh?) before giving them the new topspace.
You want more bees, more warmth and more food available - though when you give the new box you could also give syrup if the box is undrawn.
FOR NOW - I'd be worried that with a maximum 10kg of stores for the winter in their shallow box, they might be running very low. If they are "light", give them fondant by the kilo, asap.
They need lots of food available to start brooding, and you need more bees, without delay. So, make sure they don't get anywhere near starving!
And I wouldn't bother with varroa treatment until you have Q on a decent frame in a decent box. Treatment would if anything delay getting to that stage.