My experience has been that if there is a possibility of a tornado in the area, the local TV channels will warn, but they're talking to people that live in the area, not travelers unfamiliar with the area. Both times that its happened to me, I've been in KOAs. Once in "Tornado Alley" and once in southern Georgia. And both times, the owners went door to door, telling everybody.
At the one in southern Georgia, they invited everyone into the recreation room to watch TV coverage, and told us that both attached restrooms were also built as tornado shelters, including emergency generators. The owners unlocked the pop machine for us, and we all sat around telling storm stories until the all clear siren sounded. I met some nice folks and got some good CG references for where I was headed. The tornado missed us, but the next day there was damage visible from the road about 30 miles away. At least what I could see wasn't in populated areas.