In Coyne Of The Realm, an article published in the summer 2005 issue of The Call Boy, the quarterly journal of the British Music Hall Society, Peter Chorlton said that Fred Coyne (1847-84), who popularised this song, rode one daily, and for some time, and actually came on stage riding it. Velocipede (literally fast foot from the Latin) is an antiquated (and now somewhat humorous) term for a bicycle, which at the time this song was written was still in its infancy, and included all manner of odd looking contraptions, most notably the penny farthing.
Coyne's ditty contains the lines: "Everyone should try one, Everyone should buy one..."
which begs the question was he sponsored by a manufacturer? - as was George Leybourne (of "Champagne Charlie" fame).
Actually, Coyne rode a tricycle; the song has lyrics by Frank W. Green and music by Alfred Lee, and is far from unique; at this time there were many songs and pieces of music dedicated to or inspired by the new invention.
Suggestion credit: Alexander Baron - London, England, for all above