"As the 8x56 is a rimmed cartridge the corresponding 8x57 has to be the rimmed version, so headspacing is no issue. The different bullet diameter should be, 329 to 323."
We're talking about different cartridges.
The 8x56R was a rimmed cartridge for the M95 Steyr military rifles used by Austria.
That round has, to the best of my knowledge, never been loaded commercially in the United States (until very recently by Hornady) and it has never been used as a sporting round in bolt-action rifles.
That round is also NOWHERE near interchangeable with the 8x57R Mauser round.
The 8x56R round has a base diameter of .494; the 8x57R Mauser round has a base diameter of .473, the same as the rimless version.
The round that's being referenced here is, I believe, the rimless 8x56 Mannlicher cartridge which was one of the three common sporting chamberings for Mannlicher rifles, the others being 6.5 and 9mm.
The rimless Mannlicher sporting cartridges (there were several others as well as the ones I mentioned in the previous paragraph), all used cases with base diameters of .473, or the same as the 8x57 Mauser cartridge.
The 8x56 sporting cartridge was manufactured here in the US by Western, Remington, and I believe Peters through at least the 1920s.