OK, my memory has clicked in...here's the story. The songs were from a cassette tape I have entitled "David Hammond Sings Belfast Street Songs." On the cassette they were sung separately but Reiver 1 and I took these three songs and put them together as a medley. We called it "Belfast Street Songs," but that was our name for it.
A note for "I Am The Wee Falorie-Man" says "The song of a braggart traveling man. Belfast version of a song that is known to children throughout the British Isles." For "Fan-A-Winnow" the note is "A song from the linen industry -- a spinner avows her love for Barney, the band tier. Set basically to the melody of a hymn tune common in the 19th century." For "Johnny Todd" there is this note, "A simple song describing a sailor's problem when beset by his wife's infidelity." [Actually in the song the sailor and the girl are not married, and it's not exactly a "children's song" either.]
There were several other "children's songs" on the tape such as "Green Gravel, Green Gravel," "The Doffin Mistress," "The May Queen," and "King Billy Was A Gentleman."
The "Glesca Street Songs," another somewhat similar medley that we sang, came from a copied cassette tape I have of songs by Robin Hall and Jimmy MacGregor. Several of the other songs on the tape could, I think, also be considered "street songs," such as "The Barras," and "Ma Wee Gallus Bloke."