The Beatles appeared at the BBC studios Piccadilly in London on April 01 1963 in order to record two radio shows for future broadcast. Both recordings were intended for the "Side By Side" show with host John Dunn and house band the Karl Denver Trio. Unfortunately, only one of the shows has turned up on bootlegs (the second broadcast in mid-May is available and the first broadcast for late April is unavailable).
The theme song is the title of the show: "Side by Side" with vocals by John, Paul and George and the backing is the Karl Denver Trio. Songs performed for the show include "Long Tall Sally" (which at this point in the Beatles' career, had not yet been recorded officially for EMI), "A Taste Of Honey", "Chains" (both from the current at-the-time LP), the B-side to their new single "Thank You Girl" and finally ; Ringo's vocal number from the current at-the-time LP "Boys".
The tunes from the show and a bit of chat can be found on the bootleg LP "The Beatles At the Beeb Volume Two" as well as on the Great Dane box set "The Beatles - The Complete BBC Sessions".