Brand's 1960 book of "Bawdy Songs and Backroom Ballads" prints the chorus a bit differently than what he sings on the record:

"With his long john tiddlee whacker,
Overgrown kidney-cracker,
Looking for a scrimmage around the belly whang."

I still hear only the nonsensical "sidney stack" on the album. Brand's lyrics sometimes vary somewhat from appearance to appearance.

BTW, this particular jolly tinker song has not often been collected.