"Contrary to a common misconception, the shape of the mushroom cloud does not depend on the nuclear or thermonuclear component; as you note, a massive detonation of chemical explosives would produce the same effect.
"A mushroom cloud forms when an explosion creates a very hot bubble of gas. In the case of a nuclear detonation, the bomb emits a blast of x-rays, which ionize and heat the surrounding air; that hot bubble of gas is known as a fireball. The hot air is buoyant, so it quickly rises and expands. The rising cloud creates a powerful updraft which picks up dust, forming the stem of the mushroom cloud.
"The central part of fireball is hottest, creating a rolling motion as it interacts with the outer portions. Thermal instabilities, called Kelvin- Helmholtz instabilities, occur at the interface between the fireball and the neighboring cool air. If you watch a movie of a nuclear detonation, you can see entrained material swirling outwards as a result.
"All atomic bombs produce a bulge and a stem, but the really huge, flat clouds--the ones that could be described only as mushrooms-- come from the very high-yield explosions caused by thermonuclear weapons (hydrogen bombs). The fireball from an H-bomb rises so high that it hits the tropopause, the boundary between the troposphere and the stratosphere. There is a strong temperature gradient at the tropopause, which prevents the two layers of the atmosphere from mixing much. The hot bubble of the fireball initially expands and rises. By the time the bubble has risen from sea level to the tropopause, it is no longer hot enough to break through the boundary. (In other words, the bubble encounters material that has more energy than it does, so it is no longer buoyant.) At that point, the fireball flattens out; it can no longer expand upward, so it expands to the side into an exaggerated mushroom cap. The same thing happens to big summer thundercloud when they rise up to the tropopause, producing a characteristic flattened-anvil shape.
"Those interested in learning more about mushroom clouds and other atomic-bomb phenomena might want to look at the book The Effect of Nuclear Weapons, compiled and edited by Samuel Glasstone and Philip J. Dolan (U.S. Department of Defense and U.S. Department of Energy, 1977).