No, "Over The Rainbow" (1939) and "The White Cliffs of Dover" (1941) are two completely different songs written by different people. There is some irony in "White Cliffs", if Wiki is right---it was written two Americans. The Special Relationship in action.
http://en.wikipedia.org/wiki/Over_the_Rainbow
http://en.wikipedia.org/wiki/(There'...liffs_of_Dover