My understanding is the mils thing came in in the mid 1960's for most NATO countries. The reason is simple - mils are easier in terms of mental arithmetic when figuring out artillery coordinates. The original MkIII degree dials were all semi-transparent mother of pearl. This is so that the low-intensity radium lights would illuminate through the dial at night. The modern dials are not, I believe, semi-transparent. They are made of something other than mother of pearl as well and are still produced in both degrees and mills by Francis Barker who still supplies the spares to MoD. They also make dual-use dials in both degrees and mils - I have no idea if MoD uses those or not.
Given that the radium is long since worn out, I rather suspect the compasses held in MoD stores which are actual mkIII's vice later M73 compasses, have been rebuilt several times over now. Peter tells me some are still in degrees for training purposes when non-NATO countries come to train and work in degrees. Most I suspect are long since converted to mils and although the earliest M73's have mother of pearl dials, I'll bet most of the dials in use now are not the MoP type unless they are degree compasses. I really wish my compass were a MoP card version (it used to be), but the only replacement available for my cracked one was a new made francis barker degree card. Functionally that is a good thing as the old dials were only accurate to 1.5 degrees while the new cards are accurate to 0.5 degrees.
Another interesting tidbit is that not all mil cards are the same. Some are 6000mil some are 6400mil depending on which country issues the instrument. NATO countries all use either degrees (uncommon) or 6400 mils.
Without confusing things too much for ppl who don't understand mils, there are 6283.1853 mils in a circle, but the U.S. military standardised this to 6400 mils (to simplify math), so that North is seen as 6400, South is 3200. There are two pi radians in a circle, where pi is 3.1416.... One pi radian is a segment of a circle that has a rounded edge that is the same length as the straight sides and from the radian we get the mil, or mil-radian (approx. 1/1000 of a radian). Mils are now used because an angle in "x" mils is "x" units wide at a distance of 1000 of the same units. In other words, at a distance of 1000 metres, a 3 metre long object will appear to be 3 mils in length.
Since this is all based on guestimation of an object's length at a given distance, it is not a perfectly precise method, but it IS more accurate than degrees out past 500 meters or so because the unit size is much smaller. And because it only needs to be "close enough" no country actually uses a compass in 6283 mils.
Now to really confuse everyone, the Frenchuse a totally different and unique system calls the "GRAD" system where there are 400 divisions on the compass (vice 360, 6000 or 6400). I have no idea how it works, so don't ask! LOL.