It's not immediate, it is still a short time after the blast. But, at the point of the blast all of the air there is being pushed away by the explosion, not just the air specifically at ground zero. As the primary shockwave gets closer to an object, all the air between the blast and that object is being pushed out of the way. This is the initial damage you are seeing. Due to power generated by an atomic explosion, the shockwave is moving very quickly and therefor so is the air in front of it, potentially causing a lot of damage

This is actually how all bombs work. The actual blast point where the explosion and shrapnel do their thing is only a part of the damage. The rest is the resultant shockwave damage, which can also be fatal