Nick, I blew my cover years ago as I retired in 1986 and became a Chartered Planner.
From my knowledge of the Construction and Use Regs a driver need not have a spare wheel. If he does carry one it must comply with the regs. As the offence is driving.. with a faulty tyre... on a road ,the spare wheel if let say it was dangerous, there is an offence of ' dangerous parts and accessories'.
I know of one traffic pc who reported someone for only having two tyre valve covers, i.e. dangerous parts and accessories. His Sgt told him not to be so stupid as well intentioned advice was best. The driver was sent an advisory letter.