This is how much I can remember from my dad:

How well I do remember, the night that Willie died,
It was early in the morning, late at night.
Oh, the cows were singing merrily and the birds were eating hay,
And the sun and moon and stars were shining bright.

Twas the fortieth day of May when our Willie passed away,
He died harder than he ever died before.
He was sitting in a chair but he didn't like it there,
So he stood up and he died upon the floor.

We could tell that he was dead by the color of his breath,
We could hear the flowers nipping in the bud-ah.
And the doctor said the only way to save our darling boy,
Was to stop the circulation of the blood-ah.

So we filled him up with glue as we tried to bring him two,
But we only brought him eight or nine or ten.
[...can't remember how this part ends]

So we put our darling's head in a melting pot of lead,
As we lay our darling Willie down to rest.
But it was a dirty shame, for that night a burglar came,
And he stole the mustard plaster off his chest.

[It ends with the line]
That our Willie, darling Willie kicked the pail.

[So glad I stumbled across this thread. I've been wondering about the rest of the song for years. Keep it up.]