I taught "Old Bonebags" (and others from the Silver-Burdette book) to students in 1961. Some other songs included: "Barnacle Bill the Sailor", "Drill, Ye Tarriers, Drill" and many other similar songs.
Most were American folk songs, with a sprinkling of English and Irish, as I recall. (It's been 47 years, after all!)

The kid's favorite was: "In a cottage in the wood, little old man by the window stood. Saw a rabbit hopping by, knocking at his door.

Help me, Help me, Sir!, he cried - ere the hunter shoot me dead!
Come, little rabbit, come inside - safely to abide."

This was an action song that included motions and positions, by the students, to coincide with the lyrics.