Operationally, there is limited effects. No battles fought with the Krag were lost because of the inferiority of the Krag magazine system or the poor performance of the .30-40.
But some battles may have gone slightly better in Cuba and PI, so there is no rush to a new .30-03 caliber and a new rifle. So no change again in 1906, but likely the Mauser is updated with some of the features of the later 96 or 98 (shorter barrel,Double column magazine so does no protrude from the bottom of the stock, and 3rd locking lug) by time of the Great War, along with a better bullet for the cartridge itself, to a 150-170 grain Spitzer
More minor effects, no need to equip troops with the Savage P-17, since enough of the standard *Springfield have been made since 1892
So now it's 1918, and War is over. The urge for a newer flat shooting cartridge is even higher than OTL
Pedersen still develops his three cartridges. the .256, .276 and .30, all slightly more powerful that the modernized .30.40 Krag cartridge currently in service(but less than the OTL .30-06, that doesn't exist in this TL), so
it's a tossup over the .256 or .276, till something like the 1928 'Goat and Pig Board' finds the .256 far more lethal and this is adopted for rifles instead
Since it's less powerful than the .30-06, the new Rifles Trials go differently
You have the Danish 1934,M1922 and 1927 Bang Rifles
These were Gas Operated, with a cup on the muzzle, barrel not drilled. Didn't work well, despite several updates
Then the Thompson Autorifle of 1921, and Garand of 1921
The Thompson used the Blish Lock, per the SMG, and really didn't work well on .30-06 pressure(even had an oiled rounds), and the Garand still used his special primer actuated cartridges
The Pedersen Toggle Lock, and improved Garand
Pedersen used hard wax coating, while the Garand, without the primer actuation but standard primer, did not. Both rifles used ten round clips
So in this TL, You get the earlier Garand or Hatcher-Bang in .256 before the Great Depression hits
Since the .30-06 never is used for infantry rifles, the US looks for a replacement for Machine Gun cartridge at the same time
The US would probably use the French 13.2x99 cartridge for Heavy Machine guns( no .30-06 to scale up) and a 30-32 caliber round for medium MGs, as the .25 can't carry enough compound for incendiary rounds.
It wouldn't be rimmed, but otherwise hard to say what this cartridge would be, but the 300 Savage(7.62x48mm) may get adopted, more powerful than the .30-40, less than what the .30-06 would have been, and around since 1920.
So the US had a 6.5mm class Intermediate Cartridge for Rifles, and the parent of what became 308 NATO for Medium Machine guns and BAR, and .52 caliber HMG in time for WWII