I don't have any special knowledge about this image or nuclear testing, but typically when we see an asymmetry in the evolution of a system, there should be an asymmetry in the laws governing it, or the initial conditions$^†$.
There are three asymmetries that I can think of in the system that could explain the asymmetry in the evolution of the system.
First, gravity pulls down, and not in other directions. However, I'd estimate that gravity is much too weak in comparison to other forces in this system to have such a dramatic effect in so short a time, so I doubt that's the reason.
The other asymmetry that I noticed is that the shot tower is below the detonation, but not on any other sides. This seems like a plausible cause to me. The metal from the shot tower, being significantly different from the air on all other sides, may have reacted to the explosion in a very different way. Perhaps metal fragments, having significantly greater inertia than air molecules, penetrated the surrounding still air faster; the cones could be incandescent air left in the wake of those fragments.
Thirdly, it could be due to asymmetry in the construction of the bomb itself - it's hard to evaluate the likelihood of that possibility without more research.
This is based solely on some basic reasoning, though - I look forward to being proven wrong by someone with more sophisticated understanding of nuclear weapons.
$^†$ (except in some special circumstances when symmetry can be broken "spontaneously")