The phrase 'not on your Nelly' was one of the rounds in last week's radio programme Just a Minute, which prompted me to look it up. Nelly is short for Nelly duff, which is Cockney rhyming slang for puff. Here, puff means 'life; span or length of life' (OED). The OED mentions the phrase in all one's (born) puff, meaning 'in all one's life', and tends to be used in negative contexts, or with a negative verb.
Nelly and puff share a definition. They can both refer, according to the OED, to 'an effeminate man'. Puff is the older word here, dating from the very early 20th century, whilst Nelly (also written nelly) goes back to the 1930s.