There was no special British inch, etc. associated with armor thicknesses. Everybody's inch was exactly the same, at least to six or seven decimal places. The confusion arises from the 'rough' conversion of 40# being equal to 1" thickness. This was a 'rough and ready' conversion, similar to the one whereby almost everyone (in North America) calls a 1.5" x 3.5" piece of lumber a 'two-by-four' even though it is not two inches by four inches at all. Actually, a 40# plate is only about 0.98" thick. Similarly, a 1" thick plate weighs about 40.8 lbs. Close, but no cigar...
And the use of this conversion certainly doesn't mean that anybody is -- or ever was -- using "special" inches.
Armor is heavy, usually priced by the pound, and naval architects must be very careful with weights, so some navies tended to issue specifications for armor (and other plating) in pounds per square foot measure. They would therefore tend to round thicknesses off to the nearest even pounds value, e.g. 400 # plate would actually be 400/40.8 = 9.80" thick. Other users, more concerned with geometry, might let the inch thickness drive the system instead and ask for 10" thick plate. If that were the case, rough useage would convert this as 10" x 40 = 400# plate, but that's just an approximation, as the real weight would be 10 x 40.8 = 408 lbs per square foot.
Those unfamiliar with shipbuilding practice -- including the authors of many secondary textbooks -- often didn't differentiate too carefully, and often used the approximate 40# p.s.f. = 1" thick conversion as though it were strictly correct. But that's just sloppy conversion, not an example of any special inch being used.
A similar situation occurs in physics where the acceleration of gravity is equal to roughly 980 cm/sec^2. In many applications, and particularly because the acceleration of gravity actually varies a bit with altitude and position, it's 'close enough' (and makes the math easier) to set this equal to 10m/sec^2 for rough calculations. But that doesn't mean we are using any kind of special centimeter...
The correct way of approaching this is not to assume the use of some special inch for armor thicknesses, but to realize that armor plate does not actually weigh 40# per square foot, it weighs 40.8# per square foot. It's the weight that's off, not the inch-thickness measurement. If everyone did their calculations and conversions that way, there would be no problem at all.