I can't remember the exact Q plate wording, but it was along the lines of:
A vehicle made up of parts, some or all of which were not new at the time of vehicle manufacture
If it was possible to prove the age of a vehicle (or its main component parts) it was possible to get a standard age specific number plate for it, but it (IIRC) involved a fair amount of paperwork.
To get a Q plate all you needed (at least in 1990) was a valid MOT cert for the vehicle and to take it to a vehicle licencing centre who would check the chassis number and engine number before issuing a Q plate. How you got from home to and from the MOT centre, and from home to the vehicle licencing centre was a whole legal grey area
I don't know the current situation, but I believe they have tightened up the system significantly in terms of vehicle testing.