I agree with Will that the musical samples were almost certainly not left to David Attenborough. To be even more nit picking the opening recording of Estil C Ball of Rugby Virginia was not made until 1959.
The TV programmes were broadcast 1953/54. Earlier recording of Estil 1938-42 made by John and Alan Lomax were/are available but I guess that the 1959 stereo recordings being of a better quality may have influenced the producer. Not a criticism, just an observation. Likewise I believe Ian Cambell to have been a far better singer than the song's author.
I wonder if there is any chance that the radio series "As I Roved Out" could be ferretted out. If the BBC were serious about an archive of folk music surely they still have these recordings. They made it worthwhile to get out of bed and listen on a Sunday morning.

Hoot