This is from the article linked here: "In 1899, the British developed a predecessor, which was known as the .375 Flanged Nitro Express. This cartridge featured a straight-rimmed case with a length of 2.5 inches and a rim diameter of .523 inches and offered ballistics quite close to the .38/55 cartridge. This cartridge, even though almost unknown in the United States, can be encountered in rifles still in shooting condition. Even though the ammunition is no longer manufactured, anyone wanting to reload this caliber can fabricate cases from new Hornady .405 Winchester cases or use .30/40 cases. The .30/40 case ends up a little short by about .250 inches but is still very usable."
Using the 405 may require a little trimming of the rim and base diameters but probably the best choice if actual brass is not found. 303 Brit and 30-40 could be used but would be short in case length.
Keep in mind it is a nitro express about the same as the 375 NE rimless M/S, good cartridges but certainly not magnums.