Neame's parents were the photographer Elwin Neame and the actress Ivy Close. He studied at the University College School and Hurstpierpoint College. His father died in 1923, and Neame took a job with the Anglo-Persian Oil Company as an office boy. Later, through his mother's contacts in the British film industry, Neame started at Elstree Studios as a messenger boy.
He was fortunate enough to be hired as an assistant cameraman on Blackmail (1929), the first British talkie, directed by a young Alfred Hitchcock. Neame's own career as a cinematographer began with the musical comedy Happy (1933), and he continued to develop his skills in various "quota quickies" films for several years.