I can see a snippet from a book which says

"1830's under the title "All Around My Hat." The lyric was in dialect and the song bore the inscription "written by J. Ansell . . . composed and arranged by John Valentine ... as sung by Jack Reeve with the most unbounded applause"

P 383 of
The American song treasury: 100 favorites
Author Theodore Raph
Publisher Courier Dover Publications, 1986

Also:
L.C. information bulletin (Library of congress) Vol 50, 1991, p 78

The original, "written by J. Ansell, Esq." (John Hansell) and "composed and arranged by John Valentine," "as sung by Jack Reeve, with the most Unbounded Applause," pictures an English vegetable peddler, with an overloaded little donkey,


and another which claims to have the original sheet music.