I've heard of this song, but never actually sung it or heard it sung by any children or adults.

Here's two versions from http://dragon.sleepdeprived.ca/songbook/songs3/S3_63.htm

With My Hand On Myself
(actions in brackets)

With my hand on myself, vat is das here? (touch top of head)
Das is mine topnotcher, ya mama dear. (touch top of head)
Topnotcher, topnotcher, ya mama dear. (touch top of head)
Dot's vot I learned in der school, boom-boom.

With my hand on myself, vat is das here? (touch forehead)
Das is mine sweat brower, ya mama dear. (touch forehead)
Sweat brower, sweat brower, ya mama dear. (touch forehead)
Topnotcher, topnotcher, ya mama dear. (touch top of head)
Dot's vot I learned in der school, boom-boom.
Additional verses:

Eye winker - touch eyes
Horn blower - touch nose
Soup strainer - touch teeth
Lunch eater - touch mouth
Chin chowser - touch chin
Rubbernecker - touch neck
Chest protector - touch chest
Breadbasket - touch stomach
Foot stomper - touch foot
Here's another, slightly different version of this song, sent to me by Pat Dryburgh.

Put a hand on myself, and vat is duz here (point to head)
Zus is my tinkerboxer, my mamma dear
Tinkerboxer, tinkerboxer, ya, ya, ya, ya
Zat's vat we learned in the school, ya ya.

Additional verses:
Sight Seer - touch eyes
Horn blower - touch nose
Loudspeaker - touch mouth (yell "loudspeaker" for added effect)
Rubbernecker - touch neck
Gas chamber - touch stomach
Ball kicker - touch foot