meaning not on your nelly

It's a picturesque way of saying 'no'.

You can use it whenever someone suggests something that is totally unacceptable to you. It's almost archaic, so you shouldn't expect everyone to understand it.

Thank you so much...I understand.

Nelly used here is Cockney rhyming slang (London street slang) for "Nelly Duff, which signifies 'puff' which in turn means '(breath of) life'. "Not on your puff" is an earlier expression meaning "not on your life", i.e. 'emphatically not'. Various theories exist about who Nelly Duff was or whether she was even a real person. As has been noted the expression is somwhat old fashioned but may be encountered in England even today.