I assume plum (plumb?) is the US slang term meaning completely or utterly (as in plum loco, which in Westerns at any rate seems to be a colloquialism for completely insane).

The only slang usage for nelly I can think of is rather old-fashioned and means effeminate, but I have a feeling that may be a UK usage.

Plum(b) nelly should therefore mean utterly effeminate, but it seems an unlikely combination given the disparate geographic origins of its parts, and yet it's echoing around in the depths of what I laughingly call my brain as a catchphrase from somebody or other.

Bingley

Bingley