I am from London, UK, and we used to do it exactly the same way as described above, sitting in a circle with out palms facing upwards.

Lyrics:

Down by the river lives the hanky panky
Two fat frogs go from bank to banky
Zoo Zam
Zoo Zam
Down by the river goes the HONK A TONK

And on 'TONK' the child would try to slap the hand of the next one to get them 'out'.