Lina Eckenstein, in her "Comparative Studies in Nursery Rhymes," 1906, discusses the "Twelve..." and the "Gaping Wide-mouthed Waddling Frog" in her essay (on line). Both rhymes had their origin in toy-books of the 18th c. (the one with "Twelve days..." preserved at the South Kensington Museum according to Eckenstein).
Quoting Eckenstein: "The words used in playing "The Gaping..." were first printed in a toy-book of the eighteenth century (IanC gives the date and location of the book above). "Persons who are still living remember it .... as a Christmas game. "...players sat in a circle, a dialogue ensued and the answers were given in cumulative form. He who made a mistake gave a forfeit.

"Buy this of me:- What is it? The gaping wide-mouthed waddling frog.
Buy this of me:- What is it? Two pudding ends will choke a dog,
With a gaping, wide-mouthed waddling frog.
Buy this of me:- What is it? Two monkeys tied to a clog,
Two pudding ends will choke a dog, etc.
"The answer to the last question stood as follows:-

"Twelve huntsmen with horns and hounds,
Hunting over other men's grounds;
Eleven ships sailing o'er the main,
Some bound for France and some for Spain;
I wish them all safe home again;
Ten comets in the sky,
Some low and some high;
Nine peacocks in the air,
I wonder how they all came there,
I do not know and I don't care;
Eight joiners in joiner's hall
Working with their tools and all.
Seven lobsters in a dish,
As fresh as any heart could wish;
Six beetles against the wall,
Close by an old woman's apple stall;
Five puppies by our bitch Ball
Who daily for their breakfast call;
Four horses stuck in a bog;
Three monkeys tied to a clog;
Two pudding ends would choke a dog;
With a gaping wide-mouthed waddling frog."

Halliwell printed one series of answers as a separate nursery rhyme, 1842 (Halliwell's Nursery Rhymes also on line).