But The Annotated Mother Goose, Bramhall House New York mcmlxii © William S. & Cecil Barong-Gould, pp. 77-79, has the following:

There was a frog liv'd in a well,
Kitty alone, Kitty alone,
There was a frog liv'd in a well,
Kitty alone and I.
There was a frog liv'd in a well,
And a farce mouse in a mill,
Cock me cary, Kitty alone,
Kitty alone and I.

This frog he would a wooing ride,
Kitty alone, Kitty alone,
This frog he would a wooing ride,
Kitty alone and I.
This frog he would a wooing ride,
And on a snail he got astride,
Cock me cary, Kitty alone,
Kitty alone and I.

... numerous following stanzas

This has a footnote:

Much better known today is the more recent version of this old song that begins:

A frog he would a-wooing go,
Heigh ho! says Rowley,
A frog he would a-wooing go,
Whether his mother would let him or no,
With a rowley, powley, gammon and spinach,
Heigh ho! says Anthony Rowley.

The Baring-Goulds state that the original version was printed in Gammer Gurton's Garland, first printed in London in 1784. I have read somewhere that there have been efforts to prove that the later version alludes to some English notable, but I forget what that was all about.