There is a BIG difference between a watch and a warning. If there is a warning, it means that a tornado has actually been sighted in the area. Guests are told to get under cover and all outdoor rides are shut down (same thing for big thunderstorms). Some cities will sound warning alarms. But I've never been in WDW when a tornado warning was given, so don't know about alarms there.
As for watches- check any weather channel or website for the current forecast.