The tune is (or should be) “Ring around the rosie”. “Neener” is a more modern variant of “Nah nah”. “Nah nah” was used in place of the lyrics either because kids couldn’t remember the lyrics or perhaps as a more articulated hum. Search for “Ring around the rosie” and you’ll find that it’s history is rooted in nursery rhyme and possibly dancing bans, but don’t believe the stuff about the plague - it’s been well debunked.
It may be used as a song of ridicule today, but originally it was a playful taunt… essentially “you can’t catch me”.