The tour starts Wednesday with an early London premiere, moving on to Paris and ending in New York.
Blunt says it is typical of her co-star: "When I heard about the premiere idea of three in one day I just said, 'that's Tom's idea.'"
Cruise laughs, saying the 31-year-old actress from "The Devil Wears Prada" will be unconscious anyway because she's not getting much sleep with her baby daughter.
"Edge of Tomorrow" is a sci-fi action movie and love story, directed by Doug Liman. It opens in the U.S. June 6.
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