A recent performance with a folk-processed tune that seems to be getting popular:
Though the name "Black Watch" was used informally in the 18th century, it seems that the name became official - as "The Black Watch (Royal Highlanders)" - only in 1881. Greig & Duncan collected their versions in 1906.
The direct, informal diction suggests to me that the song as we know it is most likely a late 19th century creation - though of course the refrain harks back to the Napoleonic Wars.
I don't know of any broadside printings.
"But a' that they 'listed wiz forty and twa" is obviously a joke on the pre-1881 designation, "42nd Highland Regiment." Compare "The Gallant Forty-Twa," a name suggesting that there were only 42 of 'em.
The song seems not to have been well known beyond Northeast Scotland until Jeannie Robertson popularized it during the Revival.