Norman was born Monty Noserovitch in Stepney in the East End of London, the only child of Jewish parents, Annie (née Berlin) and Abraham Noserovitch, on the second night of Passover in 1928. When Norman's father was young, he travelled from Latvia to England with his mother (Norman's grandmother).
As a child during World War II, Norman was evacuated from London but later returned during the Blitz. As a young man he did national service in the RAF, where he became interested in pursuing a career in singing.
In the 1950s and early 1960s, Norman was a singer for big bands such as those of Cyril Stapleton, Stanley Black, Ted Heath, and Nat Temple. He also sang in various variety shows, sharing top billing with other singers and comedy stars such as Benny Hill, Harry Secombe, Peter Sellers, Spike Milligan, Harry Worth, Tommy Cooper, Jimmy James, Tony Hancock, Jimmy Edwards, and Max Miller. One of his songs, "False Hearted Lover", was successful internationally.
From the late 1950s, he moved from singing to composing, including songs for performers such as Cliff Richard, Tommy Steele, Count Basie and Bob Hope, and lyrics for musicals and (subsequently) films. In 1957 and 1958, he wrote lyrics for the musicals Make Me an Offer, the English language version of Irma La Douce (based on a 1956 French musical written by Alexandre Breffort and Marguerite Monnot; the English version was nominated for a Broadway Tony Award), and Expresso Bongo (which Time Out called the first rock and roll musical). Expresso Bongo, written by Wolf Mankowitz was a West End hit, and was later made into a 1960 film starring a young Cliff Richard). Later musicals include Songbook (aka The Moony Shapiro Songbook in New York), which was also nominated for a Broadway Tony and won an Ivor Novello Award; and Poppy (1982), which was also nominated for the Ivor Novello Award, and won the SWET award (renamed "the Laurence Olivier Awards" in 1984) for "Best Musical". Further film work included the theme songs for the science fiction movie, The Day the Earth Caught Fire, The Two Faces of Dr. Jekyll, and the 1963 Bob Hope movie, Call Me Bwana.
As of 2004, Norman was working on an autobiography, to be entitled A Walking Stick Full of Bagels, and musical versions of the 1954 Kingsley Amis novel, Lucky Jim, and his 1970s musical, Quick Quick Slow.