I'm exasperatedly seeking the melody behind this scene in a BBC production of Samuel Beckett's Waiting for Godot, starting here at the 1:04:18 mark. Singing more slowly, with a lilt, feels helpful. Incidentally it reminds me of Schubert's Der Müller und der Bach, but I think there's something else...
Page 125 from a book titled Beckett: Waiting for Godot by David Bradby confirms this by stating,
The tune used for the dog song was that of 'Carnival in Venice'.
Yes, I agree, the resemblance in the initial part of the melody is striking. But it's a rather simple melody (supported on very simple harmony too) that seem to me of folk/popular origin.
In the Corot production the song sounds to me like an English folks, probably a drinking song, but of course the way and language in which it is sung influence my perception.
Shubert did not usually used folk-tunes directly as source material, but folk inspired expression is much present in his melodies (see for example this excellent article).
I believe this type of melody, and even this specific melody (or parts of it), may appear in folk music from different countries. Check for example this tune from the Ozarks tradition in the US.
So though I can't identify the specific source for the Godot production, I find it more likely that both instances recur to popular/folk inspiration, than that someone in the BBC production had the idea to adapt the Shubert melody for this purpose (although it's not impossible, of course).