If You're Irish (Come Into the Parlour) was a popular Irish-American song; in Hitchcock's version of Juno and the Paycock from 1929 he uses it as a symbol of the denatured Irish culture of the Civil War years by having characters charmed by it being played on a gramophone. The sound effect, according to an article about the film, was achieved by the props man singing it in his Cockney accent while holding his nose, though perhaps this was another song. (In the same film, Sarah Allgood's stunning voice is shown off when the family and friends in the tenement room sing Believe Me If All Those Endearing Young Charms together.)

The programme does sound like the Walton's Programme. I don't remember the theme tune, but do remember the sign-off, which my parents used to mockingly repeat as "If you MUST sing, do sing an Irish song."