I think I've got most of the story off the recording, but there are still some bits I'm not sure about so please bear wth me a little longer!
It may be an effect of the old recording process but Leno's voice seems very high-pitched, with a rather affected reedy tone to it.

This is one of those convoluted comic stories about marriages that flirt with the taboo of incest. I'm sure that there are many more such songs and tales, like Billy Butler's Nell where:-

Now Nell is her mother's own mother,
And her father becomes her own son;
Her mother's first child was her father-in-law,
And her granny's the son of a gun!