Just to clarify a bit. The terms "bore", "bore diameter", and "bore size" refer to the diameter of the hole "bored" in the barrel. When rifling groves are cut, the diameter from the bottom of one groove to the bottom of the opposide groove* is called "groove size" or "groove diameter". That will also be the diameter of the bullet. In other words, the bullet is always forced through a barrel with a "bore diameter" smaller than it is, so the grooves are engraved on it.
Using inch measurements for convenience, the original 1888 8x57 rifle had a bore diameter of .311" and a groove diameter of .318". That meant a groove depth of only .0035". That turned out to be too shallow to stabilize the bullet once the barrel began to wear. The groove diameter was increased over time but that only resulted in grooves that were not fully filled by the bullet. Finally, about 1905, the German army decided to fix the problem once and for all, by going to a .323" bullet and increasing the groove diameter of the barrel to that measurement. They rebarrelled all the new Model 98 Mausers for the new dimensions, and thereafter issued only the new ammunition. Oddly enough, they called the old bullet the 8mm, and the new, larger bullet the 7.9. (As one might say, only in an army!)
The civilian name for the older cartridge was the 8x57 J (standing for "Infanterie" - the German I and J are similar and there was some confusion), and for the new cartridge the 8x57 JS, the S for the German word for large.
But the Germans had tons of Model 1888 rifles. They wanted to use them for reserves, but could not afford to rebarrel them. So what they did was just to run a new reamer into the chamber. That opened up the chamber neck, allowing the new round proper case neck expansion and there was no problem in firing the larger bullet through the smaller groove diameter barrel (remember, the BORE diameter did not change - it was still .311".
But the Model 1888 used a five round, en-bloc type clip that entered the action and fell out the bottom when empty. The Germans modified many of those to use the new 98 Mauser clip ("stripper clip" or "charger"). So, with the chamber modified and the receiver altered to use the new clip, the old rifles were considered perfectly OK and hundreds of thousands were issued to reserves in WWI and supplied to Germany's allies, mainly Turkey.
Ammunition confusion was apparently not a problem, even though Germany still had thousands of unaltered Model 1888 rifles. The new ammunition was issued in the new clips, which unaltered Model 1888's would not accept; old ammuntion was issued in the old clips, which the altered Model 1888's and Model 1898's would not accept. Loose rounds could still have been a minor problem, but in the German army's view, it was not a significant one.