This song always sounded British to me. But it actually originated in western North Carolina in the 1930s.
Its composer, John Jacob Niles, liked to take little snippets of music that people in his area sang and make them into full-fledged songs. In this case, he heard a girl singing three lines – the rest is his.
Other folk tunes that Niles wrote in this manner are “Go ‘Way From My Window,” recorded by Linda Ronstadt, and “Black Is the Color of My True Love’s Hair,” recorded by Rhiannon Giddens.
This version of “I Wonder as I Wander” is by Barbra Streisand.