An atomic explosion is in practice not an EXPLOSION at all, but an IMPLOSION. That is to say, the reaction sucks air towards the core, rather than blowing it away. The only way for the air and debris to go is up. Once the actual implosion has taken place, the shockwave then travels outwards.
... and the mushroom cloud is caused by the dust/debris being dragged up by the rising heat - the 'mushroom' shape happens as the rising cloud cools, ceases rising and is rolled outwards by the continuing updraft of debris from the point of explosion.
A nuclear explosion displaces the air, this causes an empy space, when the force of the explosion has decreased the air returns taking dust and smoke with it.
That is the mushroom.