Where severe weather is concerned, the thing that worries me the most would be hail. Very heavy rain does not phase me when I'm cycling, I'll just keep going unless it's extremely gusty as well.
Lightning, there are various strategies to avoid being hit, but with hail, stones are being thrown down with great regularity. It's harder to dodge those.
It's been suggested to me that I get a swag and in the event of hail, I get inside it then roll up-side-down so the mattress is on top of me for shelter. How do people handle hail when on tour?