The first ever episode was shown on then first of January 1964 and was presented by Jimmy "Now then, now then" Saville...The full running order for the show was as follows: The Rolling Stones with "I Wanna Be Your Man'", Dusty Springfield with "I Only Want to be With You", the Dave Clark Five with "Glad All Over", The Hollies with "Stay", The Swinging Blue Jeans with "The Hippy Hippy Shake" and The Beatles with "I Want to Hold Your Hand", that week's number one.
Incidentally, the BBC studio from which TOTP was broadcast then was in Rusholme, Manchester, between Dickenson Rd and Rusholme Grove. My Dad was a taxi driver and took none other than Jimi Hendrix to the Arosa Hotel in Withington, from my road! I'm impressed if no-one else is.