When I first read your post I thought, no chance, it went out live: As the senior member(arguably), of the Empeg site, I can remember watching it regularly and I remember the gaffes and long silences etc. of live broadcasting - in those days the whole family gathered round the set and these lapses gave us something to laugh/talk about..
As to video tapes, the BBC took delivery of their first Ampex in October 1958, so the programme(Which has been revived on a number of occasions) probably predates the record and wipe money saving era, which became prevalent by the mid 60's (Hence the missing Dad's Army episodes etc.).
However I was heartened to find two of the celebrity spots on youtube, so footage must exist: I would, therefore, certainly try contacting the BBC, but also the posters on youtube to see if they have further material.