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The London, Midland and Scottish Railway always used "LMS", never "LMR" as a short-hand reference.
The Longmoor Military Railway used a Royal Blue livery and yellow lettering and numbers - in a non-serif font, so this sign is very unlikely to be from there. See http://www.irsociety.co.uk/Archives/9/longmoor.htm for an interesting description of this line as it was in 1963, some six years before closure.