The British used the .303 round, the Americans used the .30-06. Wouldn't it have been easier for everyone to adopt the same round so they could use each others ammunition?
It is much worse than that. Even within a military, there would be multiple types of ammunition in service.
For example, the British would have the .303 for the Vickers, Bren, and Lee Enfield. The Sterling would have 9mm, the revolvers were either .455 or .38, and then the occasional Tommy might have a .45 Tommy Gun.
The Royal Armored Corps, just for its tanks, could have a 7.92mm Besa (same calibre the Germans used), or a .30 cal Browning, with a pintle .303 Bren not being unheard of (Some earlier ones used the Vickers .303 as well). The RAC were the only users in the British Army of the 7.92 round. The Vickers .50 was different to the American .50, though in fairness, the two were not in great usage at the same time.
American tanks would have 37mm, 75mm, 76mm, 3-inch and 90mm all in service at the same time, but at least everything else was some variant of .30 long, .30 short, .45 or .50. The British receiving a Sherman might remove the 75mm and replace it with a 17pr, because they didn't want the hassle of dealing with yet another calibre, the 76mm. (Any 76mm tanks went to Italy, to keep it simple, except for those the British gave to the Poles in Europe to replace their 17pr tanks). It wasn't entirely irrational: The British 75mm tank gun was designed to accept American 75mm ammunition.
If you think about it, it would be a nightmare to have to convert an entire imperial army from .303 to .30 or whatever. Whilst the weapons were being changed out, (with resultant requirements to redesign the weapons, and build ammunition for both kinds of weapon anyway), the troops at the field were using them (And generally had confidence in them). To give an example, the British attempted to re-gun all their M10s from American 3" rounds (The only system in the British military which used that 3" ammunition) to 17pr guns, which were a bit more common in the British supply system. By the time the war ended, only 2/3 of them had been converted.
Of course, after the fact, the advantages to a common standard were well recognised, and so under less pressure of current operations, the next generation of rifle and MG was all to take the 7.62mm round, 9mm for the pistol, and so on.