“Pop goes the weasel” comes from London’s Cockney dialect (I am a Cockney’s son).
“Pop” means to pawn something by taking it to one’s “uncle”, a euphemism for a pawnbroker.
My “weasel” comes from rhyming slang: “weasel and stoat” -> coat
(similarly my “trouble” is my “trouble and strife” -> wife)
The song goes:
“Up and down the City Road, in and out of the Eagle
That’s the way the money goes, pop goes the weasel”
The Eagle would be a public house wherein I spent the week’s wages, thus forcing me to pawn my coat (the only thing of value I now have) to give my “trouble” some cash to buy food!