Pop goes the weasel!
From what I could find out, ‘to pop’ was apparently slang for pawning. However there are a number of different explanations for “weasel”. One was that it stood for ‘weasel and stoat’ (Cockney rhyming slang for “coat”) and other said weasel was a corruption of ‘whistle’ and again, in Cockney rhyming slang, ‘whistle and flute’ referred to “suit”. Either way, it has something to do with a man’s jacket 
There are also opinions that the word ‘weasel’ referred to a tool used by cobblers, hatters and tailors or was something like a shuttle or bobbin, which could also be pawned.
So, the song has nothing to do with weasels popping but everything to do with going down to the pub, drinking till all your money was gone, then pawning your suit or the tools you worked with to get more money so that you could drink more.
The Eagle refers to a pub in Hackney, North London, situated on the corner of City Road and Shepherdess Walk. In 1825 the old pub was rebuilt as a music hall, which Charles Dickens used to go to, then in 1901 the hall was pulled down and the present pub built.
It’s not clear who was the author of the rhyme but one name suggested is W R Mandale.
You can find a fuller description on the origins of the song both at the World Wide Words while this site shows a picture of the Eagle pub with the verse actually written on the outside.
I have heard a reference to ferrets lacking something in their ears that keep them from getting dizzy. The thought goes that if they went around and around and got dizzy they would die, hence the “pop” in the nursery rhyme.