My great-aunt taught me this in the 50's, although I think we also sang it in the Brownies or Guides.

I remember there were different presents and the action built up.
I remember the fan (with one hand)
I think there was a rocking horse or something similar (rocking backwards and forwards while waving the imaginary fan).
It finished with a cuckoo-clock (everyone singing cuckoo, cuckoo, cuckoo, cuckoo)