I learned this with the junior section of the Orpington Junior Singers in the UK in the 1960s. The words were different:
The fiddles, they sing it and sob it and swing it,
They sway as they play it, they sing all they say.

The clarinet, the clarinet, goes doowa doowa doowa doowadet,
The clarinet, the clarinet, goes doowa doowa doowadet.

The horns, the horns, they shout it out,
The horns, the horns, they shout it out.

The drum has no trouble, just double, dub-double.
Five-one, one five; boom boom boom boom boom!

And the trumpet must trum-t-t-t-t-t-t, trum-t-t-t-t-t-t, trum-t-t-t-t,
And the trumpet must trum-t-t-t-t-t-t, trum-t-t-t-t-t-t, ta!

The bassoon it makes a point of counterpoint,
Da da da da da da, da da da da
The bassoon it makes a point of counterpoint,
Da da da da da da da.

A couple of decades later I was head of music in Homefield Boys' School, Sutton (south of London in the UK). I used The Orchestra Song but my choir seemed to be coping very well so I added some extra verses I made up myself. I wasn't aware they'd travelled anywhere; but I was very gratified to see someone mention one of mine:

The 'cello is mellow, it sings a flowing melody;
The 'cello is mellow, it sends me to sleep!

At this point my version of it slowed right down with mock snoring from the boys and stopped briefly, before they 'awoke' for a final rousing reiteration of the tutti.

The following year I squeezed yet another verse in:

The harp is so enchanting, it…
and for the life of me I can't remember what it did after that although I can remember the melody very well!

So if any of my ex pupils from Homefield remembers that one, perhaps they'd post those up here as well! I could do the score for the extra verses if anyone is interested.

And I've never heard of the piccolo verse so please post the notes somewhere!