Very interesting find. It looks to me like a 'broad arrow' in the centre, as used to indicate British government property, particularly military items, but also by other official organisations serving the Crown.
I have seen lead seals with a similar marking plus initials, like this:
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It doesn't seem to be a seal though, and neither does it seem to be a uniform button, I wonder if perhaps it's a button-type fastener from some other piece of kit, At first I thought the initials were 'RNH' for 'Royal Naval Hospital', as on this tunic button below, but with some enhancement, yours appears to be 'RNF' or 'RNE'.
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Nothing 'Royal Navy...' springs to mind and and my only other thought was 'Royal Northumberland Fusiliers'. I've never seen a button quite like it though.