I've just got my copy of Harland's "Ballads & Songs of Lancashire" (1865) down from the shelf and there isn't a version of Weaver and the Factory Maid in it.
There are two songs whose titles might be confused - "The Hand-loom Weavers' Lament" (p259) and "Hand-loom v. Power-loom" (p251), but neither seem to be a basis for the song we are looking for.
According to Roud 17771, there are several versions in the Folk Music Journal 3:3 (1977)