Copies of English Folksongs for Schools by Sharp & Baring-Gould is always available on secondhand websites even though it's a century old.
Probably get a copy for less than a quid.

If we gave you a list of suitable songs here you could easily Google and download or get some of them from the DT here. I'll start the ball rolling.

One Man went to Mow
Henry My Son
Wraggle Taggle Gypsies
Mermaid
Golden Vanity
High Germany
Three Jolly Huntsmen
Sweet Nightingale
John Barleycorn
White Cockade
Lark in the Morning
The Fox
Saucy sailor
Cuckoo
Jolly Waggoner
Dark-eyed Sailor
Frog and the Mouse
Old Woman and the Pedlar
Carrion Crow
Foolish Boy


Having got most of these from the above-mentioned book, about half of them are really unsuitable for children, being about sexual encounters or too violent or unPC.