@Dilandu has the list of questions that an Army needs to be asking in these situations.
That said, a cartridge firing a 140gr projectile at roughly 2400fps for ~2000ft-lbs energy (9g at 770m/s, 2600J) seems to be the optimum for putting a man down and out of the fight. Not too much recoil, not much need of multiple hits. 6.5mm Arisaka, .280 British, heavy loaded 6.5 Grendel...
If I am looking to feed both machine guns and infantry rifles from the same ammunition, then that's about the best option. We can argue between 6.5mm and 7.62mm for caliber, but I'd lean towards 6.5 or 6.8, depending on where you're planning on fighting. In closer terrain, the 6.8mm offers better wounding, in open terrain the 6.5mm offers flatter shooting. 7.62mm will be rather short ranged even with spitzer bullets because of the low ballistic coefficient. The downside is that 6.5mm doesn't leave much space for the interesting loads like what you'd put into a machine gun. Small AP cores, not much space for tracers, etc.
If I can design my own stuff without caring about what else is out there, I'd want a longer cartridge than the 5.56x45 or 6.5 Grendel in rifles and MGs. Don't have to go as long as the 7.62x51, but that long 9g bullet is taking up space in the cartridge case that you would prefer to fill with powder. Also, things like tracers or AP cores are less dense than lead, so an AP bullet would be longer than a plain FMJ. Wouldn't take much, maybe 3mm more overall length with the same case volume. So call it 6.8x39mm with an 11.2mm base diameter (yes, good old M43 case as the foundation, because it's just big enough), case overall length of 60.5mm (6.5 Grendel case but 6.8mm bullets and 3mm longer OAL).
If we are not feeding rifles and MGs from the same ammunition caliber, then we can discuss going smaller on the individual rifles, and/or keeping the 6.8x39 in the rifles and going larger on the MGs. I personally would not want to go much smaller than 6mm for the individual rifle projectile.
For going bigger on the MGs, I'm quite a fan of .338 caliber, though the .338 Norma is better for magazine or belt fed weapons than Lapua. This wouldn't be for every belt fed in the army, though. The squad automatics in each fireteam would be something low recoil like that Surefire MGX or a Knights LAMG, in the rifle caliber. The platoon tripod MGs and vehicle MGs would be 8.6x63mm, and built to fire for greatly extended periods (like the difference between an M60 and an FN MAG).
For pistols, I might actually get weird and go with .40S&W. Loading that to .45acp weights and speeds gives tolerable recoil with good terminal effect, and would let you build a PDW/SMG for your rear echelon troops that works like the FN P90. .40S&W (and 10mm) are very straight walled cases, so would fit into that style of magazine without trouble. More recoil than the 5.7 version, of course. Otherwise, I'd mess with 5.7x28 or similar as the pistol and rear echelon troop weapon caliber. Not totally happy with the 5.7, to be honest, because of how long it is. My Five-seveN was a reach to the trigger, even though the grip was nice and narrow.