I was thinking maybe you'd get some kind of generalised alert that tornadoes may be expected overnight, but generally, you'd be safe. If there are sirens anyway, it doesn't sound like you'd need the phone on.

If I (bizarrely) got a message informing me that a tornado was about to hit my house, I'd a) assume it was a scam b) have no idea about what I was supposed to do about it anyway. Hold on to something solid to avoid getting sucked away?