It seems that planning for the car began in 1943 as the tide of war in Europe was turning toward the Allies. Styling is credited to Veniamin Samoilov who seems to have been aware of American and other Western design trends prior to Hitler's invasion of Russia in June of 1941.
Styling the Pobeda was easier than producing it. As the link above indicates, a few were built as early as 1946, but it wasn't until 1949 that mass-production was achieved. Production in Russia ended in 1958.