Indeed, that was how I learned it as well.
I learned it as a small child but
I'm not certain if it was at school, in fact,
or one of the songs in
my parents' highly eccentric collection of
recordings of traditional music and singers.

The hee hee hee bit, I mean.
I do notice, from looking at the other posts and links,
that the all-American version linked to in this thread,
says
"you and me"
rather than
"hee hee hee" --

interesting, that "you and me" was not in the version I learned.


Finally, there is one instance from my childhood from
which I have a recollection of "Little Brown Jug"
here in the United States;
however, it was the tune alone, not those lyrics.

This instance was in elementary school: one of the activities
introduced to us from an early age
was some form of dancing, I'm trying to recall
if this was physical education
or music class, and I've blurred the two.
What I recall was that on this one occasion,
we were introduced to what was called the Virginia Reel.
The soundpart, including the caller and calls,
were all one pre-recorded package:
the instructor put the needle on the spinning vinyl disc,

(... yes, I KNOW that dates me ...)

and the recording hollered at us:
"HONO[U]R YOUR PARTNER!"
and off we went.

The musicians on the recording struck up the tunes,
and the caller went on hollering over the music.
And whilst he hollered, I recall taking in the instrumental dance music.
It was a pastiche, or pot-pourri, or whatever one calls it:
many tunes all mixed together.

And about the time that the couples,
one couple at a time,
started going down the row and then coming back,
I recall hearing the banjo break out:
"Ha ha ha,
you and me,
Little Brown Jug,
How I love thee ..."   

and the ensemble chugged on down the tracks,
to mix my metaphors.
Thanks for listening, sorry to ramble too much.