"Not on your Nellie!" is a rather definite negative statement in English, but who is this Nellie?

Alas, it appears that there isn't (or wasn't) an actual lady called Nellie. The expression is a short version of an example of rhyming slang. This site explains:


http://www.worldwidewords.org/articles/nellie.htm

Another curious expression my father used back in the 1940s was not on your Nellie, which some authorities think had been imported from the USA about ten years beforehand. Like Charlie it sounds home-grown, to the extent that it has been suggested that it is actually a piece of London-based rhyming slang: “Nellie” = “Nellie Duff” = “puff”. There was certainly an older slang phrase in existence: not on your puff, meaning “not on your life; never” in which “puff” means “breath” and so “breath of life” and so life itself.

To suggest that it was in fact rhyming slang needs some explanation of this supposed person Nellie Duff. Now duff has a number of senses. One of them appears in plum duff where it is just a regional pronunciation of dough. Another sense is “useless; rubbish; counterfeit”, once a common British word, and one which was taken to the US by Scottish settlers. So it is probable that the two mildly disparaging words were put together just to make the rhyme. This happened with another defunct expression Nellie Bligh for “fly” (the name of the much-traduced captain of the Bounty had by the late 1930s become a conventional term for someone tyrannical or deeply unpleasant, no doubt helped along by Charles Laughton). It may also have been linked for its inventors with memories of a late nineteenth-century slangy rhyming couplet: “Did he marry poor Nell? / Did he hell!”.

I’d love to find that there really had been a Nellie Duff and that there could be an opportunity to ask her shade (for she surely must by now have passed to her heavenly reward) what she thought of her name. If it were me, I’d have changed it, quick.

http://www.worldwidewords.org/articles/nellie.htm

I found a bit more information. It is a trunkated expression that meant "not likely" because it was based on the traditional song "Nelly the Elephant" and the full phrase was an expression about being about as likely as riding into town on somebody else's elephant.

http://www.phrases.org.uk/bulletin_board/20/messages/181.html

I can't get into Mudcat, but from Nellie's English Bookstore (of Japan!) here's a correct version (unlike the strange spellings and verse shapes you get whenever the Toy Dolls are involved). There is no riding of Nellie into town, or anything like, except for her leading the grand parade:
Verse 1

To Bombay
A traveling circus came
They brought an intelligent elephant
And Nellie was her name
One dark night
She slipped her iron chain
And off she ran to Hindustan
And was never seen again

Verse 2

Nellie the Elephant packed her trunk
And said goodbye to the circus
Off she went with a trumpety-trump
Trump, trump, trump
Nellie the Elephant packed her trunk
And trundled back to the jungle
Off she went with a trumpety-trump
Trump, trump, trump
Verse 3

Night by night
She danced to the circus band
When Nellie was leading the big parade
She looked so proud and grand
No more tricks
For Nellie to perform
They taught her how to take a bow
And she took the crowd by storm

Verse 4

The head of the herd was calling
Far, far away
They met one night in the silver light
On the road to Mandalay
So Nellie the Elephant packed her trunk
And said goodbye to the circus
Off she went with a trumpety-trum
Trump, trump, trump

https://genius.com/Toy-dolls-nelly-the-elephant-lyrics