In 1963, Noël Coward created the part of the fish and chips peddler "Ada Cockle" specifically for O'Shea in his Broadway musical, The Girl Who Came to Supper. Her performance of traditional Cockney tunes charmed the critics and helped win her a Tony Award for Best Featured Actress in a Musical.
In 1963, O'Shea was a guest on The Ed Sullivan Show. She was popular enough that she came back in 1964 and shared the billing with the Beatles. Their joint appearance drew what was then the largest audience in the history of American television, helping bring her to American audiences. She was a member of the repertory company on the short-lived CBS variety show The Entertainers (1964–65). In 1968, O'Shea was cast in the television movie The Strange Case of Dr. Jekyll and Mr. Hyde, which earned her an Emmy Award nomination for Outstanding Performance by an Actress in a Supporting Role in a Drama.
