The Old Woman Who Swallowed a Fly is undoubtedly related to several other old women of the British Isles.

There was an old woman who lived in a lamp.
She hadn't no room to beetle her champ*
So she up with her beetle and broke the lamp
And then she had room to beetle her champ.

There was an old woman who lived in a sink.
She lived upon nothing but victuals and drink.
Victuals and drink were the whole of her diet
And this poor old woman could never be quiet.

And related nursery rhymes.
"Sometimes in the spring our potatoes were peeled and this was to make champ. Champ was a child's delight, potatoes lightly mashed (beetled) with hot milk, seasoned with salt and pepper and scallions, served in big soup plates.... One summer I was down at Ballycarry for the haymaking and I was told that one allowed a quarter stone of potatoes for the champ and a pint of buttermilk for each haymaker."
Alice Kane, ed. Edith Fowke, 1983, "Songs and Sayings of an Ulster Childhood," p. 40.

It is likely that the old woman (lady) rhymes were at home in England as well as Ireland, and that they are quite old.

There was an old lady of Rumford
And she was a gay old lass
And many an honest penny got
By selling asperagrass-

Beginning of "Old Woman of Rumford," Bodleian Library, Harding B16 (187a), 1813-1838, J. Catnach. First four lines. This song (and similar) about an old woman and her daughter may be the adult inspiration for the children's rhymes. "There Was an Old Woman and She Was Blind" is another.