IIRC it had a little row of wooden pipes standing vertically which worked in much the same manner as fipple flutes (recorders), rather than the metal-reed system used by harmoniums. Those are very much older: you see them in medieval iconography, often sitting in the player's lap. They're also called portative organs or positive organs, and the keys were often played with one hand while the other worked the bellows; though there were also slightly larger ones which had someone standing around the back doing this chore. (I used to play with an early music group that had one.) So flute organs are actually mini pipe-organs, and their sound is slightly less edgy.

I saw Dolly play hers in gigs with Shirley a couple of times (and what a magnificent pair they were) but I can't remember how she handled the bellows issue. Sometimes people just use a little fan-motor, unfolky as it sounds.