Barbara, you asked (wow, 16 years ago!) what other rounds get paired with "Ah poor bird," "Hey ho nobody home," and "Rose." In my family, we often added (to a similar tune that diverges the farthest in the final phrase):

We have gathered
from the hills and factories
Trying to understand
this great wide world.

Interestingly, I see no mention of this one anywhere on Mudcat or in a quick Google search, and I know nothing of its origins!