The 8x56R cartridge was only developed in 1930 as a way to keep the old rifles in service. Austria and Hungary both had been hit very hard by the Great War, as had been Bulgaria, the only other general user of these rifles.
Austria adopted the cartridge in 1930, Hungary in 1931; it is likely that conversions began immediately.
Once the remaining rifles had been converted, Hungary adopted a new rifle in 1935, using the same 8x56R cartridge. This was the 35M, which was later produced for Germany in 8x57 and termed the Gew. 98/40 in German service, the 43M in Hungarian service. It was a standard 4-motion turnbolt Mannlicher-type rifle with 2-piece woodwork, the 8x57 rifles being redesigned to use Mauser chargers.
Bulgaria, the only other country having a national supply of the straight-pull Mannlichers, followed suit, converting their supply of M.1903 rifles.
Hope this helps.