Adam Boulton on Sky has just said that Dimbleby will be moderating the debate on one side of the university before dashing across to the other side to present QT. From that I take it they have separate studios.
The Daily Mail will be sharpening their pencils then - not only have they spent money on one set, they've spent money on another too!
I know you said that with tongue in cheek somewhat, but Question Time is something of a travelling circus - it brings with It its own set, its own crew, its own OB trucks.
When it's been at TV Centre, Mentorn (the indie who make it) dry-hire a studio, and Arqiva park a scanner outside - and as far as I know, use their own cameras. The only difference is that they plug their output into TVC rather than doing a satellite uplink.