I don't think there's any contest on this. For all the whooplah about the AK-47, the 8x57JS was the primary round for everything from Maxims to Mausers, MG-42's to Stukas. On BOTH fronts through BOTH world wars. Total killed in those wars mounted over 100 million, and a fair chunk were killed by Central/Axis troops using small arms firing the 8x57JS. The 7.62x54R, 30'06, .303, and other assorted cartridges of the same wars also did their share of combat but obviously only on certain fronts, not both fronts.
In contrast, the post-war conflicts involving the AK-47 from Vietnam on down have been much lower intensity and have involved far fewer deaths from the 7.62x39 round. Still a lot, but not close to the Mauser's round.
I would say:
1--8x57JS
2--7.62x54R (from Imperial Russian conflicts to Afghanistan)
3--6.5x50 Arisaka (the Chinese invasion was unspeakable slaughter using this round)
4--7.62x39 (most post-1965 conflicts)
5--.223/5.56 NATO (ditto, and we've dished out better than we've taken with it)
6--.303 British (colonial conflicts and both WW's)
7--.30'06 (key service during part of WWI and part of WWII, but typically on offensive only)
8--.308/7.62 NATO (multiple early cold war conflicts plus third world use)
9--.22 LR (extensive criminal and low-intensity conflict use since mid-19th century. VERY old)
10--9x19 (ubiquitous since WWII with a history back to WWI, used in pistols and SMG's)