Poor Omie/ Little Omie Wise (or Omie Bell, Poor Naomi and so on) is usually killed by her boyfriend John Lewis rather than by her mother.  That song seems to be modelled on the Oxford Girl/ Witham Miller series of songs.  The only Little Nonie that I can find any reference to was a song written by Gordon Waller and Sharon Sheeley in the late 1960s, and released in 1967 (Capitol 5886, in the US) as the b side of a single, Speak For Me.  That's the Gordon of "Peter and Gordon", incidentally, who described themselves as "folksingers", which meant "singer-songwriters", really, though I believe they did once record a set of Barbara Allan.  Mind you, so did everybody else.