To the same tune as "Turkey in the Straw" is another song, much more extensively self-referential, called "Old Zip Coon":

Oh, there was a man with a double chin,
Who performed with skill upon a violin,
And he played in time, and he played in tune,
But he never played anything but "Old Zip Coon".
"Old Zip Coon" he played all day,
Till he drove his friends away, etc.

For all we know, the man with the double chin may himself have been old Zip Coon.

Kipling's "Absent Minded Beggar" mentions "Rule Britannia" and "God Save the Queen".