I have to admit, you made me go check my references. According to Canfield and others, the Remington 1903 for Britain project never made it beyond the paper stage. Production was never started. Perhaps a toolroom sample or two were made, but I've never seen any photos or other documentation of these. Do YOU have any?
I kinda think this is awfully confusing to poor old salty, he's asking questions while we're debating the angel/pinhead question!

Serial on the 1903 is on the front receiver ring. The 8x57 (8mm Mauser) serial should be on the side of the receiver ring, or on the side rail. The 8x57 is a Mauser, different from your 1903.