Apart from anything else, this is an interesting song from a dating point of view. The subject matter means that providing earliest and latest dates for it is much easier than it is for most traditional songs. Wikipedia provides a good summary history ...

History

The word punch is a loanword from Hindi panch (meaning five) and the drink was originally made with five ingredients: alcohol, sugar, lemon, water, and tea or spices. The original drink was named paantsch.

The drink was brought to England from India by sailors and employees of the British East India Company in the early seventeenth century. From there it was introduced into other European countries.

The term punch was first recorded in British documents in 1632. At the time, most punches were of the Wassail type made with a wine or brandy base. But around 1655, Jamaican rum came into use and the 'modern' punch was born. By 1671, documents make references to punch houses.

From this, and the evidence above, we can conclude that the song could not have been written before 1655 and must have been written by 1844 when Pitts certainly had it in his catalogue.

I think we can get closer than this as I have, at home, a more detailed catalogue of the broadside publishers' output.

:-)