This English folk song, also known as "Dashing Away With The Smoothing Iron" or simply "The Smoothing Iron," appears to originate from Somerset, and as far as can be ascertained was first published by the musicologist Cecil Sharp (1859-1924), who also arranged it. Although most definitely a song, it can also be considered a nursery rhyme.
Unlike many folk songs, the words - though about a mundane subject - are not inferior to the melody. The British Library catalogue lists several arrangements including for brass band and full orchestra. It is definitely not singles material, but was recorded by Manfred Mann as "Dashing Away With The Smoothing Iron" for the B Side of "My Little Red Book." This version has a slightly jazzy feel to it, but completely misses the mark.