I wonder if the familiar kids' song somehow evolved from this one:

From The Columbian Songster: Being a Large Collection of Fashionable Songs for Gentlemen & Ladies (printed by Nathaniel Heaton, Jun., 1799), page 61:

[No title is given]

The dusky night rides down the sky,
And ushers in the morn.
The hounds all join the jovial cry.
The huntsman winds his horn.

[CHORUS]
And a hunting we will go,
A hunting we will go,
A hunting we will go,
A hunting we will go.

The wife around her husband throws
Her arms to make him stay.
My dear, it rains, it hails, it blows,
You cannot hunt to day.

Yet a hunting, &c.

Sly reynard now like light'ning flies,
And sweeps across the vale,
But when the hounds too near he spies,
He drops his bushy tail.

Then a hunting, &c.

Fond echo seems to like the sport,
And join the jovial cry.
The woods and hills the sound retort,
And music fills the sky.

Then a hunting, &c.

At last his strength to faintness worn,
Poor Reynard ceases flight.
Then hungry homeward we return
To feast away the night.

And a drinking, &c.

Ye jovial hunters in the morn
Prepare then for the chace.
Rise at the sounding of the horn,
And health with sport embrace.

When a hunting, &c.