I would be more inclined to think it was in Beligium since WWI or at least was out of the U.K. military system in the WWI time period for the following reasons:
1) It has the magazine cut-off, that was pulled for land service in what, 1915/1916? If it was in the BritishArmy or RAF in 1939/1940, why would it have the cut-off still in place? That seems to imply to me it was lost relatively early in WWI
2) It has the lobbing sights. Now a lot of the SMLE rifles sold off in the US of A between 1958 and 1968 came out of the big British surplus sales and most of the SMLE MK III with 1914, 1915, and early 1916 dates seem to have been reworked, and so marked on the rear wood. All of these rifles have the lobbing sights removed that I have seen that also have the commercial proofs applied (i.e. they came out of British UK stocks and not from some other source such a Belgium, Portugal or Ireland). The rework mark I refer to is the Star and E on the underside of the buttstock.
3) I have never seen a picture of any British troops in the 1939/40 time frame equipped with SMLE MK I variants. Also I doubt that there were sufficient spares for these rifle in the system in 1940, as when the Irish requested parts to update their rifles in 1940/41, no MK I spares were supplied, at least according the limited documentation I have found relating to forends and safeties supplied to the Irish in late 1941/early 1942.
None of the above is in any way proof positive, but the alternative is that a SMLE MK I with WWI features (lobbing sites, magazine cut-off) was never put through the WWI/ post WWI rework but managed to remain in circa 1915 issue features, issued in WWII and lost. The number of 1914 and 1915 SMLE MK III rifles in the US of A that are in original shape is truly small, as in virtually unknown. 1916, 1917 and 1918 yes, but not the early dates. It makes a lot more sense that it left the British military system sometime in the WWI period.
So you ask why would it be placed in a barn around 1940? Well one of the first things the Germans did in 1940 was to publish a number of flyers that anyone found in the possession of arms would be shot. That part of Belgium was not occupied by the Germans in WWI; it was in the area where the British armies were stationed. A fellow that had found the arm around WWI might have thought it was prudent to stash it someplace other than his house once the Germans arrived.