Looks like this one has a distinguished history. Here's the information from The South Riding Tune Book for which I suspect that Malcolm Douglas is responsible.

The Oxford Book of Nursery Rhymes (Iona and Peter Opie, 1951) has the following entry:

Polly put the kettle on,
Polly put the kettle on,
Polly put the kettle on,
We'll all have tea.

Sukey take it off again,
Sukey take it off again,
Sukey take it off again,
They've all gone away.

"Chappell in Popular Music of the Olden Time says 'The well-known country dance and nursery song, Polly Put the Kettle On, was transformed into a Scotch tune for Johnson's Scots Musical Museum in 1797. This was about three years after Polly had become very popular with the young ladies by means of Dale's variations for the pianoforte.' It was hardly, however, a question of transformation. The tune had long been known as Jenny's Bawbee, which was mentioned by Herd in 1776, and given by Joshua Campbell in 1778. Possibly the words were old English. The rhyming of tea with away is as in Pope's The Rape of the Lock (1714), though it still survives in some country districts. Polly was a common pet-form of Mary, as was Sukey of Susan, in middle-class families in the mid-eighteenth century. Grip, the raven in Barnaby Rudge, when very much exited cried, 'Hurray! Polly put the ket-tle on, we'll all have tea; Polly put the ket-tle on, we'll all have tea. Hurrah! hurrah! hurrah! ' Around 1870 'Polly put the kettle on and we'll all have tea' was a much repeated catchphrase.

Old Nurse's Book, Charles Bennett, 1858:

Sukey take it off again,
It will all boil away.

With additional verse

Blow the fire and make the toast
Put the muffins down to roast,
Blow the fire and make the toast
We'll all have tea.

Sheffield Glossary, S.O. Addy, 1888:

Come Betty, set the kettle on,
Let's have a cup of tay;
Sukey take it off again,
We'll have no more today."

Polly Put the Kettle On
Text from a broadside published between 1840 and 1866 by C. Sheard of 192, High Holborn, London.

I am a merry, happy chap,
A jolly cove am I,
One of those very merry boys,
Who never will say die;
I have a very loving wife,
Her age is twenty-one,
She's like me and other folks,
So full of harmless fun.
Sometimes we invite a friend,
An hour or two we pass,
For nothing beats at home,
A social pipe and glass;
And when the clock is striking -bum!
It's then I sing with glee,
"Polly put the kettle on,
We'll all have tea."
Polly put the kettle on,
The kettle on, the kettle on,
Polly put the kettle on,
And make it sing with glee,
Polly put the kettle on,
The kettle on, the kettle on,
Polly put the kettle on,
We'll all have tea.

Now when a very little lad,
I've often heard them say,
Polly put the kettle on,
And clear the things away;
Don't sit reading there so long,
But get the tea things laid,
And when being done,
A pretty aspect made.
There's nothing looks much nicer,
On a cold and frosty day,
Than to see the toast and butter
Shining on the tray;
So like I've heard it when a child,
Those words will stick to me,
Polly put the kettle on,
We'll all have tea.
Polly put the kettle on, &c.

A brother I've a sailor,
The other day came home,
He told us of his travels,
And where he'd been to roam;
He told us of his battles,
What wondrous things he did,
And then he'd hoist his slacks,
And chew another quid.
I gazed with all astonishment,
At his altered style,
And judge of my surprise,
Although it made me smile,
For with a voice of thunder,
To my missus shouted out,
"Polly put the kettle on,
We'll all have tea."
Polly put the kettle on, &c.

No matter wherever I go,
Or when invited out,
As the clock is striking four,
You'll hear my welcome shout,
Some folks may laugh at me,
For them I do not care.
As I'm a chap as you can see,
I'm jolly everywhere;
So not to break the friendship,
I ask you one and all,
Some afternoon when you have time
On me to make a call,
And as the clock is striking four,
It's then I'll shout with glee,
"Polly put the kettle on,
We'll all have tea."
Polly put the kettle on, &c.

:-)