Wolcott was born in Flint, Michigan, USA. He moved to Hollywood in 1937 and soon began working at Walt Disney Studios writing music for cartoon shorts, then feature films such as Pinocchio and Bambi. By 1944, he had become General Musical Director at Disney Studios. In 1950, he transferred to MGM Studios as Associate General Musical Director, and in 1958 became General Musical Director.
Wolcott had US hit singles in 1944: "Tico-Tico", and 1960: "Ruby Duby Du".