Of course "wrong tune" makes sense in this context - it's a historical context, not a "folk context" whatever that may mean. If the programme was showing people singing in the 1850s, using a tune composed more than a century later, it was wrong.

The song people in northern England would have been singing in the mid-19th century would have been "Good night and joy be with you all", in one of the versions by Scott, Hogg or Alexander Boswell, and using the tune current around 1800 (which was reproduced so many times without significant variation that the idea of a "folk process" applying to it back then is nuts). They would NOT have been singing Dominic Behan's take on a twentieth-century Irish version of the words, and they would not have been using an unrelated tune.