I agree about the first few bars on the basis of the version Brian posted. Maybe this is why in the article they said 'slight resemblance'. After that not so much resemblance. Problem is so many folk tunes resemble each other, as Julia Bishop points out.

I can't access the version of 'Unfortunate Rake' in Holden, mentioned in Kidson & Broadwood's FMJ article, but I did find the other one they cite, in 'Crosby's Irish Music Repository'. This indeed resembles the song tune only slightly. However, the version from 'Merrie Melodies' copied out by Broadwood, resembles it much more strongly.