[1] Born in London, Shaw played with Teddy Foster's band during World War II at the age of 15.
He visited the United States in 1947 with close friend and fellow pioneer bebopper altoist Freddy Syer, then moved to Canada after he was unable to secure a work permit.
He was one of the early Club Eleven players, along with Ronnie Scott, John Dankworth, Lennie Bush, and others; he also played with many of these musicians on the recordings of Alan Dean's Beboppers.
After Club Eleven shuttered, Shaw played with Vic Lewis and toured Europe with Cab Kaye, then joined Jack Parnell's ensemble in 1953 and Ronnie Scott's nonet in 1954.
Shaw played regularly both live and as a session musician for many British jazz musicians over the course of the next twenty or so years, working with Joe Harriott, Tony Crombie, Don Rendell, Tony Kinsey, Stan Tracey, Bill Le Sage, and others.