The states cooperate with the federal on many things on a voluntary basis. I cannot imagine any state bringing suit to block participation in a national public safety exercise. After all, national defense is a major reason we have a federal system in the first place, is it not?
I am old enough to remember these kinds of tests being conducted during the cold war, a long blast of an irritating tone, followed by the usual “if this had been a real emergency ...” message. In fact, I knew a disk jockey who lost his job by jokingly revising the message as “if this had been a real emergency, y’all would’ve been blown to bits by now.” Some people just have no sense of humor.