Im not sure if warnings are issued for severe storms, but its a good bet that if there is lightning, then the big rides and any outdoor ride are closed. I think they wait 20 minutes to re-open the ride after the last bit of lightning is seen.
I was there during a pretty nasty storm, lots of lightning, rain, and wind, but everyone just stayed in the shops or restaurants. I do know that everywhere in Disney has emergency plans and safe places specifically for tornados and tornado warnings. Ive never been there when anything like that has had to be used though, but all of the parks, as well as DTD have these plans.
I dont think Disneys ever had a tornado
or even a tornado warning really, but they are plenty use to hurricanes, youre going a little early for those though, they usually hit September-ish. And even with hurricanes, only with the most severe do they close the parks.
If you ever get concerned though, ask any cm with a radio. If the weathers bad, they track it pretty much continuously so that they can have an idea on ride times. Theyll be able to tell you if the storm is looking to last longer, or if its looking to get worse.