Billy Weeks very kindly says go ahead so I'll post the sheet music details first. It is a 2-page folded sheet typical of late 18thc, with flourishing script titles. (seraph for s at the start of words)
The Babes in the Wood
Written and composed, to introduce an
Admired Cry in the Town of Leicester &c.

London, Printed by Cabusac & Son's, No196, Strand.
for Henry Valentine, Leicester. Pr. 6d/.
(The publisher details give us a dating window of 1794-8)

key of A 6:8

The Babes in the Wood! The Babes in the Wood!
Don't you remeber the Babes in the Wood?
When a Child on the knee How silent I'd be,
While my mother related the Story to me
Of the Babes in the Wood, The Babes in the Wood,
Don't you remember the Babes in the Wood?

2
My Dear, you must know,
That a long Time ago,
There were two little Children, whose names I don't know,
Who were stolen away,
On a fine Summer's Day,
And left in a Wood, as I've heard the Folks say.
Poor Babes,&c.
3
And when it grew Night,
O sad was their Plight,
the sun it had set, and the Moon gave no Light:
They sobb'd and they sigh'd,
And bitterly cry'd;
Then, poor little Things, they lay down and died.
Poor Babes,&c.
4
A Robin so red,
When he saw them lie dead,
Brought Strawberry Leaves, and over them spread:
Then all the Day long,
The Branches among,
He'd prettily whistle, and this was his Song --
Poor Babes,&c.

There follows a repeat of the first verse with melody for 'Guittar'
and then the melody for 'Ger. Flute'

Billy also went online and found the following extract.
Quoted from Wm. Gardiner 'Music and Friends, or Pleasant Recollections of a Dilletante' 1853, which is about Gardiner's musical life, mainly in Leicester. He was born in 1770.

'Another early production was the following little song, from a street cry that much interested the children of Leicester. An itinerant vendor of toys with a musical and plaintive voice, paraded the streets with 2 little wax figures in a bower, representing The Babes in the Wood. To this morceau I persuaded Mr Thomas Combe to write some lines, of which I made the song that was published'.

h
He then gives the same words and music as the above sheet music.