No worries, it's always nice to have a look at these old photos.
I think the Sun, on closer inspection (i.e. having had a proper look) might actually have the VTS/Vitesse engine - the barrel and exhaust aren't right for the Villiers engine.
In terms of finding out details related to the number-plates, we're slightly in pot-luck territory here.
In the 1970s, before such records were really prized, and certainly prior to any useful form of bulk media storage, local authorities were ordered to destroy all such records.
Luckily for us, some didn't for whatever reason, and they're what survives today.
Your best bet is the Kithead Trust, which holds some records itself, and gives a list of where others are held.
The info they give for Sheffield is as follows:
SHEFFIELD
Sheffield Archives, 52 Shoreham Street, Sheffield S1 4SP (0114 273 4756)
Registers 1903-05 (W 1 - W 413) and 1931-77 (WJ to DWA 729N)
Unfortunately, it looks as though the later W series no longer survives.
Others might though - the list of holdings for England and Wales is to be found here:
http://www.kitheadtrust.org.uk/england_wales2_11.htmlYou can match a prefix letter (or letters) to an issuing authority here:
http://www.kitheadtrust.org.uk/new_page_13.htmlNick