Bumping this thread in light of a current thread (containing a Christmas-themed parody of "Donkey Riding") in which a poster states that the song neither refers to literally riding a donkey nor to a donkey engine.

I am presently doubtful that the phrase does not plainly refer to riding a donkey and I'm curious to know when the idea of the donkey engine might have creeped into discourse.

So, I'm adding a few additional sources that haven't appeared (fully) yet here.

Wood, Thomas, ed. The Oxford Song Book. Vol. II. Oxford University Press, 1927.

Page 63 has "Donkey Riding" with score.

Were you ever in Quebec
Stowing timber on the deck?
Where there's a King with a golden crown
Riding on a donkey.

Chorus:
Hey! ho! away we go
Donkey riding, donkey riding
Hey! ho! away we go,
Riding on a donkey.


Were you ever off the Horn
Where it's always fine and warm,
And seen the lion and the unicorn
Riding on a donkey?

(Chorus)

Were you ever in Cardiff Bay?
Where the folks all shout 'Hooray!
Here comes John with his three months' pay
Riding on a donkey!'

(Chorus)


Notes: "...was sung thirty or forty years ago [1887-1897] on Lancashire ships and schooners which ran from Liverpool and Glasson Dock to Canada for timber... I am indebted to Mr. Walter Raby for this song."