Depp, 50, and Heard, 27, met on the set of their film "The Rum Diary" in 2009 and announced last year they are dating.
Depp broke up with French singer and actress Vanessa Paradis, the mother of his son and daughter, in 2012.
Directed by Matthew Cullen, "London Fields" co-stars Billy Bob Thornton, Jim Sturgess, Cara Delevingne and Theo James. It is to be released next year.
Notable deaths of 2014 [PHOTOS]