Wilson was born in Sale, Greater Manchester and was educated at Harrow School and Oriel College, Oxford. During the war he served in the Royal Army Ordnance Corps in Great Britain, Egypt and Iraq. While at Oxford he wrote revues for the Oxford University Experimental Theatre Club and then attended the Old Vic Theatre School on a production course.
Most of his work for the stage was material for revues, such as Hermione Gingold's Slings and Arrows, Laurier Lister's Oranges and Lemons, and See You Later, starring such performers as Peter Cook. He wrote the book, music and lyrics for The Boy Friend for the Players' Theatre in 1953. Its success resulted in a longer version being produced in the West End at Wyndhams Theatre. After its opening in January 1954, over 2,000 performances were put on there. It opened on Broadway in 1954, at the Royale Theater, and introduced Julie Andrews in her Broadway debut. The show ran on Broadway for over 480 performances.
Wilson wrote the musical Valmouth in 1958, based on a Ronald Firbank novel set in a seaside resort. In 1964 he wrote Divorce Me, Darling!, a sequel to The Boy Friend.
He died in Taunton, England in 2014, aged 90. He donated his papers to the Harry Ransom Humanities Research Center at the University of Texas at Austin. His autobiography, published in 1975, is titled I Could Be Happy.