Theoretically, every atom splitting or fusing is a ‘nuclear explosion’. I assume you mean with a bomb, meaning an uncontrolled chain reaction. In that case, the W54 artillery projectile (Better known as the “Davey Crockett”) is probably the smallest, with an adjustable yield from 1kt to as low as .01 kt. That’s 10 tons of TNT, which is only about twice as powerful as the Amphol bomb used by Timothy McVeigh in the Oklahoma bombing. Of course, there are radiation problems with a nuclear bomb, but still, this is pretty small. One could go off across the street from you and you could survive it.
One niggling point: There’s nothing special about a nuclear bomb in creating a ‘mushroom cloud’. Mushroom clouds form when energy is released in a large enough amount that the changing atmosphere itself affects the shape of the cloud.
In a large nuclear explosion, the initial blast forces a hot column of air upwards. As it rises, adiabatic expansion eventually causes it to cool and condense, creating a cloud. At first, thermal pressure from below causes it to shoot upwards, creating the stem of the cloud. Eventually, the bubble of hot gas slows down and starts to expand, again from adiabatic expansion. But because it’s no longer being forced upwards, it turns into a regular cloud, which is the ‘cap’ on the mushroom. If it travels high enough, it will encounter the jetstream, which will shear the top off, creating an ‘anvil’ shape familiar to anyone who’s seen a giant prairie thunderstorm.
If you could make a nuclear explosion the size of a firecracker, you’d get a firecracker-type cloud, which would differ only in the way the residue in the types of materials in the bomb is presented (much like different fireworks shells make different explosions based on the composition of the powder and impurities inserted into the powder to control burn temperature, color, etc).
So, while I don’t know exactly what a small nuclear explosion would look like, we know it wouldn’t create a room-sized mushroom cloud.