Neame was born on 23 April 1911 in London, England. He studied at University College School and at Hurstpierpoint College. Neame was married to Beryl Heanly from 1933 until they divorced in 1992. Then he was married to Donna Friedberg from 1993 until his death. With Heanly he had a son, Christopher. Neame also had three grandchildren. Neame died on June 16, 2010 in Los Angeles, California from complications of an accidental fall, aged 99.