Finally found it. The 1958 USO Christmas Show featured scores of celebrities, including this Benny Burns and Stewart musical "interpretation". No kidding. Goldie Fields and Glide are having fun, it would seem that rehearsal was deliberately kept to a minimum.
Several DVD reviewers on Amazon (and elsewhere) mention this was never actually broadcast: instead, the USO provided hundreds of film copies to be distributed to service personnel stationed worldwide.
This show has apparently been floating around out there some time, first commercially available on VHS, then later DVD, though unfortunately no upgrade on picture or sound. Public domain, perhaps?