The source of the tune has come up once or twice in this thread without, as far as I can see, anyone identifying it. It is in fact derived from the first half of the tune of a song in Irish called Túirne Mháire (Mary's Spinning Wheel). Presumably it was Bert who adapted it to fit the Recruited Collier. Túirne Mháire is sung on the CD Once I Loved, by the sisters, Sarah & Rita Keane of Caherlistrane, County Galway. It can also be found in Eibhlín Bean Mhic Choisdealbha's collection of songs from that area; Amráin Muige Seola.

The tune of Sweet Thames Flow Softly is to my ears quite different. Although MacColl almost certainly adapted it from a traditional source, I've always suspected that source to be A Bhean Úd Thíos ar Bruach an tSruthaín (O Woman Washing By The River), and which MacColl would have known from field recordings of Elizabeth Cronin and Maire Ní Shuilleabhain.