Hi, long time lurker/first time poster here.
I joined up to reply to this question, as I think can shed some light on it.
Most companies will only insure Q's 3rd party only, without fire and theft cover, but some will give you the F+T cover you want. I insured my old Q plated Escort with Swindle (sorry, Swinton), and they were perfectly ok about it. You could also try taking out a traders policy assuming it doesn't omit Q platers.

Once a Q, always a Q.....in that the vehicle has to wear a Q plate for the rest of it's life. Most companies will ask why it's been Q'd too, so make sure you know. Also, remember that the emmissions test part of the MOT is not applicable to Q platers. The only check is a visible one, ie look to see if the car is "smoking". If not, it's a pass.

Hope this helps.