Although I said it was the words that caught my interest, it may be that the tunes just may be a help.
First off "Black Cat Piddled.." (Aus tune), "The Girl I left Behind Me" and "Jim Along Josie" are all polkas (although you can switch a polka and a reel fairly easily.) The Australian tune begins rather like "Old Dan Tucker". It is nothing like "The Girl I left Behind Me" or "Jim Along Josie".
You can easily sing the black cat verse to any of these songs.

I'd be for a minstrel connection for all of them as they occur in Australia.

Many tunes changed titles here, either with other well-known tunes or else they were re-named completely. Musicians got bored with words when they played for dances and used floating verses -- as they did also in America.

Further to Azzizi's comment about London -- Note that London was a "hot spot" for 19th century performers.
Cheers, Joy