One of our morris side sings it to the tune of Villikins and his Dinah (Sweet Betsy from Pike to our transatlantic cousins) and the chorus is:

10 9 8 7 6 5 4 3 2 1
10 9 8 7 6 5 4 3 2 1
10 9 8 7 6 5 4 3 2 1
10 9 8 7 6 5 4 3 2 1

The first verse is similar to yours, and the second goes:

Down on the farm I was milking a cow
But how to milk it I didn't know how
I pulled its tail instead of its tit
And all that I got was a bucket of
10 9 8 7 6 5 4 3 2 1 etc

Those are the only two verses he does, and if it has a title I suppose it would be "10 9 8 7 6 5 4 3 2 1"

Wassail! V