From the liner notes of my copy of Tommy Sands' Singing of the Times:

Humpty Dumpty fell off the wall... was the first "fact" that I learned as a child. If all the "truths" that I was inoculated with since, through education, press and media are anything to go by, then Mr. Dumpty probably was pushed off the wall. This was first performed "live" as part of a play with the same name at Queen's University Festival in Belfast.