The source of the bullets and rifle is irrelevant. The problem is the person targeting the British soldiers. Proving that the same rifle was used in each case means it is likely that there is ONE person to track down.
The rifling pattern might be enough to make an educated guess about the manufacturer of the rifle. Rifling can spiral in a right or left direction. The rifling can be created by cutting, button rifling, or by forging the barrel around a mandrel. Each manufacturer has a particular technique. S&W uses a "gain twist" rifling that rotates faster at the muzzle than at the breech. The value in that approach is less deforming of the bullet.
One sniper. Using 5.56 NATO Lake City manufacture ammo. Like others have said here, they themselves have bought surplus 5.56 Lake City, so the ammo may or may not have come from directly from currently used U.S. forces, supply source.
Or it may have, since we've provided more than a few of our own M-16 variants, and ammo, to Iraqi forces. Some of it, once under the authority of the Iraqis, has walked off. Or so news articles I have read [a while back] have reported. I have no reason to believe that this is not possible.
This sniper might be toast already, if he'd been one of Mookie's boys.