I think the EAS would fire off but likely after the fact. As 9/11 showed, TV was on it right away. Radio grabbed TV audio, then went to their own originated material, generally via a news network. I suspect firing off an EAS National Activation takes some form of protocol delaying it for a minute or two, maybe longer. I suspect media might beat EAS to the punch in reaching some, while the EAS would grab the rest.
Some say the EAS should have been fired off on 9/11. I won't speculate either way. A national emergency, yes, but media likely moved quicker than the EAS could have in most places. Even in the sticks somebody would have seen the TV reports and called a local station or communicated in some way.
Funny thing, I was doing sales in radio when the Gulf War began. We simulcast an ABC TV station local cast and swiftly got permission to carry their audio. Then my boss called CNN after seeing their coverage minutes later and we became a CNN News affiliate instantly, switching to their feed.