More references to "stuck in the mud like Billy Barlow". One has "stuck in the mud was Billy Barlow".
Another has, "I'm fairly stuck in the mud. Billy Barlow" This one is from an article from Portland, Victoria, Australia in 1846.
This may indicate that a Billy Barlow song with a line "stuck in the mud..." was well known by this early date.
Still haven't found the song but it seems I need to keep looking before 1846 in English and American collections.
Cheers, Joy