The second version whose chorus Steffan quotes above as sung by John McCutcheon is called 'Here's to Cheshire', a fairly recent re-write by Cheshire fruit farmer Leslie Haworth and recorded by The Spinners in 1964. I haven't checked Dicho's links but I think it unlikely this is among them, so here goes:

HERE'S TO CHESHIRE
(Trad / Leslie Haworth)

Chorus:
Here's to Cheshire, here's to the cheese
Here's to the pears and the apple trees
And here's to the lovely strawberries
Ding! Dang! Dong! go the wedding bells

There was an old frog lived in a well
Ding! Dang! Dong! go the wedding bells
A pretty little mouse lived in a mill
Ding! Dang! Dong! go the wedding bells

Froggy went a courting and he did ride
He said 'Missy Mouse you must decide'

For I am rich and I am brave
What better husband could you have

I'll not take your word for that
You'll have to satisfy Uncle Rat

Rat said, I'm very much afraid
If you don't take Froggy then you'll die an old maid

So the knot was tied secure and fast
She's off her uncle's hands at last

Open the oysters spill champagne
Never will there be such a feast again

Tune up the fiddle and let's have a square
Top couple must be the happy pair

As they were going it hot and strong
The old grey cat come prowling along

Into the window and out of the yard
Didn't bring no invitation card

Uncle Rat like a hero stood
Puss wet her whiskers in his blood

Miss Mouse she made a dive for a crack
Puss made a pounce and broke her back

Where was the valiant frog this while
Just about broke the four minute mile

Came to a brook and he didn't stop to look
The old drake knabbed him in his fluke

That was the end of him and of her
Won't be no tadpoles covered in fur

Lines 2 and 4 are invariably 'Ding! Dang! Dong! go the wedding bells'. I've left them out for reasons of space.