Bseed: The Church's One Foundation tune is best known as Fred Karno's Army:
We are Fred Karno's Army, the ragtime infantry
We cannot march, we cannot fight, what bloody use are we?

(Fred Karno was a music-hall comedian.)

As regards the Dog song, I am confident that Alan Bennett used it in Forty Years On. I seem to remember that the children sang it at Speech Day while the teachers sang Foundation. But it's a long time since I saw it and I may be wrong.