Pop, to pawn, goes back a ways, appearing in Francis Grose, A Classical Dictionary of the Vulgar Tongue, London, 1785 (noted in the main thread on the song, 19247- link below).

However, as shown in the lyrics and explanations posted in thread 19247: Pop Goes the Weasel

the origin is probably erotic (see post in that thread by Masato Sakurai, comment by Fuld).

The meaning to pawn possibly took over in the late 18th c. versions.
In the time of Victoria and Albert, it was a movement in a party dance. The childrens' songs are all later than the 1850s.

In America, of course, most people think it has reference to the actions of the animal.