> It appears that the only letter that does not appear anywhere in a registration is “Q”.

"Q" is used for certain types of motor vehicle in the UK - certainly cars home-built from kits, and (I think) vehicles of the ASP variety - extensive restorations where so many bits from other cars have been used that one cannot readily identify the completed car with any of its donors.

Bingley is perfectly correct about "I" and "O" being omitted from the prefix (or suffix,if you go back far enough), although they do, of course, appear in the main registration letters. "Z" appears in the main registration if the car was registered in Northern Ireland (Ulster/The Six Counties, depending on your political persuasion) but I think you are right in saying that it doesn't appear as a pre/suffix.

Thank goodness the essential vowels do appear in the main registration, otherwise the ability for ingenious souls to produce personalised and amusing number plates would be severely circumscribed.

My favourite was an old Triumph Herald with the reg.
"TEN 66"