I have 15 inch wheels running 185/55 tyres. The cycle wings and rear arches I have only just fit.

What's the official line on how much of the wheel has to be actually under the arch ?

The short answer is, I don't know, but I may have an idea!
Years ago, I was on my way from Somerset up to Santa Pod, towing a car I'd built, on an A frame, behind a Sierra Ghia, when Mr Plod pulled me over.
He was very cocky, and told me I was breaking the law by using the A frame without a lighting and brake connection to the towed car. I pointed out the car weighed less than the weight limit requiring lights and brakes. He was confused by what I'd told him, and he obviously didn't know the law as he thought he did.
He was visibly livid, and was determined to get me for something, so after lots of looking and prodding, he almost yelled "Ah-ha!".
He took a strip of wood from his car, and stood it vertically (ish) beside one of the rear tyres, and said the tyres were outside the guard, and therefore were illegal.
I produced a builders square from my tool chest, and set his stick square.
Sorry for the long drivel, but I think the tyres would have to be covered by the guard for probably something like 50 degrees either side of TDC.

You have to pity the poor motorist who was pulled over straight after you!!

I think you will find that the tread must be covered sideways. Otherwise I don't think there is a requirement for the amount of circumference to be covered.

Is this man allowed in 'ere ? ('es a brave bugger I'll give 'im that !)

As I understand it, there should be no more than one inch of tread protruding outside of wheel arch (on a vertical plane that is). This was back when the humble Mini was the boy racer car of choice and wide wheels and spats were the order of the day.
Things may have changed though.

Terry
Showing my age I think!