As James J. Fuld says (in The Book of World-Famous Music), "The weasel was originally a metal tool used by hatmakers in England which was popped (i.e., pawned). In the American words, however, the weasel is unmistakenly an animal." See the cover of this edition (Buffalo: J. Sage and Sons, 1854).

There's an entry in Eric Partridge's Dictionary of Slang and Unconventional English, vol. 1:

pop goes the weasel!, now gen. regarded as a nursery-rhyme tag, was in the 1870's and 80's a proletarian (mostly Cockney) c.p. [catch-phrase]. Ware, 'Activity is suggestive by "pop", and the little weasel is very active. Probably erotic origin. Chiefly associated with these lines--Up and down the City Road / In and out the Eagle, / That's the way the money goes, / Pop goes the weasel!

However, according to quotations in O.E.D. (1st ed.; s.v. Pop, int., adv.), the suggestive meaning apparently disappeared when it was danced at court balls.

c1854 (Music-seller's Advt. in Newspaper), The new country dance 'Pop goes the weasel', introduced by her Majesty Queen Victoria. -- Musical Bouquet No. 409, Pop goes the Weasel; La Tempête; and Le Grand Père. These fashionable dances as performed at the Court balls. 1855 in N. & Q. 10th Ser. IV. 211/1 This dance is very popular, it it without deception, 'Pop goes the weasel' has been to Court, and met a good reception.

~Masato