I think that design is a bit far fetched but I think the premise has promise.
OK how about this. The US adopts Pedersen's design at a very early stage in development. Garand, who if memory serves was Canadian, looks for alternate buyers for his design, first in his native Canada, who express an interest, and then the British who are intrigued by what the Canadians have discovered. Britain adapts the Garand for their uses. i can't see the brits using an en bloc clip so what we have here is a rifle that's old fashioned in some ways (using a high powered rimmed cartridge) but cutting edge in others (the first detachable box magazine rifle in service to be used as such). The SLG (Self Loading Garand, pronounced "Slag") gets adopted in 1938 (same year as the Lee Enfield No.4)
Anyway, the US use the .267 round for their rifle and develop a carbine version for tankies etc (thing analogue to the M1 Carbine), which is later adapted for a larger magazine and full automatic fire, giving the US a proto assault rifle at about the same time as the StG 44.
Post war, the .276 (or a modernisation thereof) becomes the standard NATO round. Plus points for britain is that it's easier and cheaper to rechamber to a smaller round, and so the EM-2 and Taden projects go ahead, meanining that the British armament indusy is in a stonger position than OTL post war from selling indegenous designs to commonwealth nations. Given the fact that the British are switching to a Canadian designed weapon pre WWII, this could lead to closer Commonwealth cooperation during the war.
Instead of the Bren getting adopted the British army decides to do a Charltonesque conversion of the old No.3 bolt action rifles. The GaSter (GArand STERling) automatic rifle similar to that pictured above, gets tabled as an LMG/Automatic Rifle for tank and commando units around about 1941 but later becomes popular enough to replaces the pseudo charltons by around 1943.
Come Nato standardisation the Automatic rifle could be rechambered and a new mark with a drum feed ordered as an section level LSW.