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that 1960's map that I linked to is quite unusual in using a scale of 1:63360 (1 inch to the mile) but the the grid overlay is in 1km squares.
Nothing unusual about that, as any of us agéd ex-rallymen know!
The Ordnance Survey used 1":1mile until the change to the 1:50,000 scale in 1976 (I think).
Military users needed a way to define a position on the map and at some stage the WO Cassini Grid was introduced, which was a grid of kilometres. That was still in use when some maps I have were printed in 1942.
That seems to have been refined into the National Grid by the time my 1947 maps were printed and of course it's still in use