One theory goes that the black ash created by a nuclear explosion would cool temperatures at the Earth’s surface, resulting in less temperature difference between the lower and upper atmosphere. Rainfall would drastically decline and thrust large regions across the globe into drought.
But I'm more inclined to believe that God will cause the draught like in Elijah's days.
'The skies over your head will be copper, and the earth beneath you, iron. 24 Jehovah will make the rain of your land powder and dust that will come down on you from the heavens until you have been annihilated." (Deu.28:23,24)