I think that the "Q" plates signify a newly registered vehicle, but one that is built from older parts for which no registration survives. "Q" Plates are also given to non-standard vehicles or special builds that are not modelled on existing manufactured vehicles.
You have to submit the vehicle for a Ministry of Transpoert Vehicle Inspectorate examination for roadworthyness before the plate is granted.
I built a Mini-Cooper from bits I obtained mostly in scrapyards. I used all the proprietory parts with no modification. A body from here, an engine from there, some subframes from BMC, wheels from dismantlers, and so on. I obtained a log book with the body and kept that registration. I submitted it for test and having passed (lots more strict than the yearly MOT!) I had the logbook stamped and off I went. Legal car.