The UK inch size is the diameter (not easy to manage and depends on direction !). Google "UK Hat size" and it will give you a link to circumferance measured above the ear.
There was I think only ever one shell size for the helmets and it is possible to alter the size by removing some of the horse hair padding,
I have successfully altered mine to fit.
The steel BMB DR helmets have 2 sizes of outside shells. Up to 7 1/4 it has the smaller size and from 7 1/2 and up it has the larger outside shell.
The best way to find out which size is right is just try one on.
The liners were made in 1/4-inch size increments, from Size 6 & 1/4 through to 7 & 3/4 inch........sizes 7 & 1/2 and 7 & 3/4 used a larger-size shell..............
Trying-on for fit is the best method, but if it helps a modern 56/57cm size equates to the old Size 7, a 58/59cm a 7 & 1/4, a 60/61cm a Size 7 & 1/2 and a 62/63cm a Size 7 & 3/4..............whilst there were lots of these on the market a number of years ago now, today anything above a Size 7 is hard to find with the larger sizes difficult and expensive to source...........