Here's an early version of The Old Sow for comparison. It is definitely the ancestor of The Old Sow but whether it has given rise to the song here is anyone's guess.
It comes from the second edition of The Whim of the Day, printed at Drury Lane, London, in 1792, p16.

The Little Pig
A Favourite Song
Sung by Mr. Fawcett, in the Mayor of Garrat.

The little pig laid without any straw,
Haugh (snorts) ha paugh ha (squeaks like a pig) ha shaw dau diddle dau;
The little pig laid without any straw,
Lingigo, lara lang, O Susan's a little woman.

We thought this pig it would make good pork
Haugh, hork paugh hork, hork shaw dau diddle dork
.....Susan she's a little woman.

I thought this pig it would make better bacon,
Haugh acon paugh acon, acon shaw dau diddle dacon etc.

At last we kill'd this little pig and of his hair we made a wig,
Haugh, hig paugh hig (tohu) (whistles) hig (squeaks) hig shaw dig diddle dig,
At last we kill'd this little pig O, etc.

It could be that this rather graphic song was softened for polite children's consumption, but my grandfather sang the graphic version to us kids and we were delighted.

'Favourite' is an accurate description, the song continued to be printed throughout the 19th century, and was revived in the 1920s by various yokel type singers, to achieve a second wave of popularity.