There was an old lady who swallowed a fly is cumulative nonsense song. There were several versions known by the 1940s, but the well-known version, recorded by Burl Ives in 1953 was written (and copyrighted) by Rose Bonne and Alan Mills the year before.
For further information see: https://en.wikipedia.org/wiki/There_Was_an_Old_Lady_Who_Swallowed_a_Fly