Its the 5.56mm vs .223 all over again. If you have a 5.56 chamber you can shoot both, the Wyle chamber is designed for both, but if you have a .223 chamber, you can only shoot .223. IIRC, that the NATO case has a few dimension changes that are so tiny, there should be no problem feeding and extracting, its possible that if you had the shortest head spaced .223 chamber at the end of the tolerances and the longest 5.56mm case at the end of the tolerances, you may run out of headspace, and the bolt might not go into battery, but that would be very rare, most chambers are cut to have a little extra space than the end of the tolerance. The problem is pressures and its not because one is loaded a lot hotter than the other (the spec pressures are different because they are measured differently between SAMMI and NATO, the actual pressure are pretty close to the same). The problem is the bullet set back and the length of the throat in the chamber, the .223 chamber with a 5.56 in it, as the bullet hits the end of the throat because of the differences in lengths, causes a pressure spike higher than the .223 chamber is rated. But, at the same time, the pressure spike is still within the safety margins required for the chamber, that is why so many people get away with it, even though you're not suppose to use 5.56 in a .223 chamber.
Just looking it, since I don't have a 7.62 Rifle yet. Seems the same is true between .308 and 7.62, except .308 does have more pressure and its not supported properly in the 7.62 chamber to handle the pressure. So, its the opposite of 5.56/.223, you want the .308 chamber to be able to use both .308 and 7.62. And the chances are less likely you'll get away with it, if you use .308 in a 7.62 chamber, unless its a modern 7.62 chamber cut for both.