On the other hand, if we are looking at adopting the .270 or .280 British (or similar 6.8mm-7mm round) then really we are accepting a compromise to behave as a universal round for use in the standard infantry rifles, assault rifles, and LMGs. In those circumstances the .270/6.8mm is probably the better choice. The .280 itself was a compromise proposed by the British to appease the US requirement for long range shooting (taken to the extreme of compromise with the .280/30). Modern evaluations and ballistics development have revealed that a longer and heavier .270 FMJ (in the 115gr - 130gr range, making for a very long bullet with high cross-sectional density) produces just about the perfect balance of ideal trajectory, stability, and wounding capability. As I stated in an earlier post, a mid-length .270 round--something like the wildcat .270-08, perhaps even a shade shorter like a 6.8x48mm (longer and more powerful than the modern 6.8 SPC) instead of 6.8x51mm--would probably be the best option as it allows a relatively short action for high rate of fire while also permitting 1000 yard accuracy equal or superior to the 7.62x51 NATO of OTL with the right load.
The issue with that is that while the 'ideal' round is good at 1000m, you're just making a better WW1 rifle caliber...which was basically already perfected with the Swedish 6.5mm round. Great for sniper rifles and MMGs (even 'heavy' mgs), but that only creates an overkill infantry hand weapon round. Because for infantry hand weapons in the era before optics could be placed on all infantry weapons, combat was largely limited to 300m with maybe 7% happening above that, because even if the terrain was flat you still couldn't really see someone in drag color clothing trying to make use of cover beyond that point. That's also before the smoke and other sight impediments of battle kick in. With combat at 300m or less then you've got yourself a long range round, which is difficult to control in automatic fire, doesn't permit as quick of follow up shots (assuming no muzzle brake), and you can carry fewer of them than a smaller caliber. Given that the VAST majority of rounds fired are misses (something like 25,000 rounds fired per casualties inflicted in WW2) or more probably suppressive fire, you're wasting the 'ideal round' not even hitting anyone to use it's wounding abilities. It is to keep their head down while you maneuver on them, call in fire support, or can use your squad/platoon intrinsic explosive weapons to kill, wound, or route them. At that point you're probably worse off than even using an intermediate 7.62 round. Jack of all trades is a master of none. As the US found out with it's combat experience there is simply no replacement for the 5.56, either smaller or larger, that is worth the cost of replacing it, because it fits it's niche so well, even with it's penetration/wounding issues (fixed now with the M855a1 round it seems). Range issues are something that would be solved by using a heavier caliber for select special weapons that can actually use it that have scopes or ability to fire a large volume of bullets with tracers to see where the bullets are landing at a distance. A ballistically better option already existing with the Swedish 6.5mm, which could be modified to a smaller case by using more modern powders, for direct fire while IMHO indirect fire was better done with weapons other than rifle caliber MG so a 'HMG' at the battalion level as in WW2 shouldn't exist....or if it did be something in a caliber like the 9x66mm MAS or the .338 Lapua used in the LWMMG...something that is able to penetrate against fixed positions or vehicles while being very long range without being as heavy or powerful as the .50 Cal M2.
Neither option really offers much, truth be told. The smaller calibers (6mm-6.5mm) will offer lighter cartridge weight (possibly also permitted a lighter rifle/weapon system) and increased control under automatic fire at the expense of "stopping power" similar to where NATO settled with the 5.54x45. It is because of this short coming that it may desirable or even required to keep a heavier .30+ caliber round for long range and heavy hitting requirements between that light infantry round and the 12.7x99mm (.50 BMG).
Regarding the .280, it also doesn't offer much in terms of universality that the 7.62x51mm doesn't also do, which is why I was advocating for a .270 caliber round which best balances the advantages of the smaller .22-.26 caliber rounds (light weight, low recoil, flat trajectories) and the .28-.32 caliber rounds (stopping power, momentum, resistance to drift/deflection, armor penetration at range, etc.). I guess, the summary is that if you are willing to accept a two caliber arsenal then take one from the lower end and one from the higher end--as NATO has had since the introduction of 5.54 to pair with the 7.62--where as if you are really serious about a "one-size fits all" universal round the best compromise is some in the 6.5mm-7mm range, with 6.8mm being the best balance of all. I was basing it on the premise of NATO not accepting 7.62x51mm as per the OP.
Even with something like say the 6.5mm round stopping power isn't going to be a big problem, but you will need a 'double version' standard, that is a short ranged round optimized for infantry hand weapons and one for MGs and sniper rifles like the German or Soviet intermediate and full power versions.
You'd probably be better off in terms of having LMGs/MMGs and sniper weapons, but with a not-so-optimal infantry hand weapon, plus lack of interchangeability between the long and shorter range rounds anyway. You'd really need a stubby and probably unbalanced 6.5mm bullet pushed as fast as possible to get that tremendous wounding effect the 5.56mm is known for (and apparently the 'Pig Board' discovered when they tried out the 6.5-7mm rounds for wounding ability on pigs; they also found that none had a super advantage over the others within 300m, but the 6.5mm round was the most damaging). They found the 6.5mm flat based bullet pushed to at least 800mps did the most damage, so I'd imagine if you got a 85 grain 6.5mm round to 850-900mps wounding power would be just fine. Recoil should be close to a .224 Valkyrie round (that is more powerful than the most powerful 5.56 NATO round, but significantly less than a 7.62x39 round), so you'd be good on that, but the kicker is not being able to carry many of them and having limited range performance due to designing the round for best wounding ability within 300m rather than aerodynamics for ranges beyond that.
All that said I don't get why the US in trying for 1000m performance optimization for the 7.62 NATO round didn't recognize the advantages of the 6.5mm round in that role other than desiring to minimize cost when adopting a new round by trying to keep the existing caliber and just modernize the cartridge.
What is even more inexplicable is the Brits doing the same with the .270-.280 rounds. They knew the 6.5mm round had the best ballistics and that the Swedes already basically perfected the long range bullets in that caliber if they were going to go for distance performance (the Swedish 140 grain 6.5mm bullet vastly outperformed the British 7mm/.280 140 grain bullet in distance performance) and given the US experience testing the wounding ability of the various calibers would have know that a 6.5mm round would wound better than a 7mm round within 300m (infantry combat distances), while retaining the best velocity out to long ranges if they wanted a 2000m MMG round (which they were trying to get the .280 round to do...without a boat tail for some reason).