Neither of the above, I should think. Although some later broadside texts have 'note', others have 'nut' (this is mentioned in the Chainmaker notes), as do some examples from oral tradition.

These are supposed to be impossible tasks, so 'nut' seems more likely; though it isn't a particularly impressive image even by broadside hack standards. In the discussion at http://www.folkinfo.org/forum/topic.php?topicid=306, someone quoted (in all seriousness) a rather more extravagant mishearing:

'Go fetch me a Nun from the dungeon deep'

Now, there is an image to conjure with.

I don't doubt that George Dunn sang 'note'; whether he had any particular meaning in mind I wouldn't know (he doesn't seem to have commented on that detail; perhaps nobody asked him). On the whole I'd think that's just how his father sang it, and that neither of them worried about it.