How much ionising radiation is released during an atomic blast?
The amount of energy in an atomic blast releases as radiation (gamma rays, beta particles, alpha particles and neutrons) depends on how the bond is designed but isn't usually a big proportion of the total energy released. But that's not why we worry about the radiation from a bomb.

The reason why we worry about radiation is the radioactive products created in the blast (eg from  fission of the large nuclei in a fission bomb or from the potentially radioactive nuclei created when the initial radiation blast interacts with other materials). Typically many dangerous isotopes are generated that continue to release radiation for days or years. Worse the explosive blast scatters many of these over a large area.

But not all bombs are the same. Neutron bombs are designed to release far more of their energy as radiation (in this case a large blast of high energy neutrons) with the express purpose of killing people in the immediate area while having a much smaller blast. The original idea was to be able to use them on invading Russians in Western Europe without doing too much damage to the infrastructure. Neutron bombs can release ~40% of their energy as neutron radiation which is far more than the 5% released in a bomb tuned to create a large physical explosion.