A couple of weeks ago I was at the range and the noob next to me was shooting an AR. He was new to shooting. One of the local gun shops had just sold him a high end AR chambered for 308 and a few thousand rounds of 7.62 and 308 ammo. A lot of it looked like reloads but the noob didn't have a clue whether it was factory or reloaded ammo.
He showed me a mangled 7.62 round (the case was crimped, dented, scraped and the bullet had been pushed about 1/4" into the case and was canted at about a 7 degree angle. He told me he had a jam and he had tried to chamber another round (the mangled one) on top of the cartridge that had failed to eject from the chamber. Scared the crap out of me. Every time he tried to fire a NATO round he got another jam. The 308s feed just fine.
I know most ARs chambered for 308 are supposed to shoot the NATO rounds just fine. However, some 308 chambers don't like some NATO rounds.
While this may work for modern firearms, older firearms may be more finicky.
I've always gone by the "5.56x45 chambers will feed 5.56 and .223, but not all .223 chambers will feed 5.56" - which is borne out by the SAAMI list.. This could be due to chamber pressure being a bit higher for the mil-spec rounds. There is apparently also a feed ramp angle difference, which can cause .223 chambers to have issues feeding 5.56.
9mm NATO rounds may be loaded to different chamber specifications, which might be a factor with some (particularly older) 9mm Luger/Para guns.