Does a nuclear explosion mushroom revolve?

What do you mean by "revolve"?

All explosions, nuclear or conventional, produce a "mushroom cloud" its just that the yield of a nuclear explosion is so much higher that makes the cloud bigger, more obvious, and more persistent.

The mushroom cloud is formed by convection currents from the heat of the fireball: the stem is created by the updraft of the fireball rising through the cool atmosphere, the cap is a toroidal convection cell with the fireball at the center - air is drawn in at the bottom, heated by the fireball, rises above the fireball and on encountering cool air again cools and spreads forming the top of the cap, eventually it spreads enough that there is no more hot air rising under it to hold it up and it falls forming the sides of the cap, usually some of this falling air gets sucked back in and reheated by the fireball as it gets to the bottom of the cap.