The OP was not completely mistaken. Although Down Below is a Sidney Carter song, it is also one of a number of Carter songs promoted and performed live by Flanders and Swann. For example:

"The Youth of the Heart" would be sung and played by Swann as a solo during the "At the Drop of a Hat" tour.

"Down Below" was recorded live on tour in Canada, and the recording released on Flanders and Swann's "Hat-Trick" compact-disc set, I think on compact disc number 2 titled "Hats Around the World." As with "The Youth of the Heart," Flanders is absent, and Donald Swann sings and plays "Down Below" as a solo.

Together, Flanders and Swann would sing Carter's "Good Literature" with its refrain line, "I'm waiting for the film to come!"

Swann also performed, again as a solo, "Take me Back to Byker," affecting a Newcastle accent.