Energy of a fission nuclear bomb comes from the gravitational energy of the stars.
Protons and neutrons can coalesce into different kinds of bound states. We call these states atomic nuclei. The ones with the same number of protons are called isotopes, the ones with different number are nuclei of atoms of different kinds.
There are many possible different stable states (that is, stable nuclei), with different number of nucleons and different binding energies. However there are also some general tendencies for the specific binding energy per one nucleon (proton or neutron) in the nuclei. States of simple nuclei (like hidrogen or helium) have the lowest specific nucleon binding energy amongst all elements, but the higher is the atomic number, the higher the specific energy gets. However, for the very heavy nuclei the specific binding energy starts to drop again.
Here is a graph that sums it up:
http://en.wikipedia.org/wiki/File:Binding_energy_curve_-_common_isotopes.svg
It means that when nucleons are in the medium-atomic number nuclei, they have the highest possible binding energy. When they sit in very light elements (hidrogen) or very heavy ones (uranium), they have weaker binding. Thus, one can say that for the low "every-day" temperatures, the very heavy elements (like the very light ones) are quasistable in a sense.
Fission bomb effectively "lets" the very heavy atomic nuclei (plutonium, or uranium) to resettle to the atoms with lower number of nucleons, that is, with higher bound energies. The released binding energy difference makes the notorious effect. In terms of the graph cited above, it corresponds to nucleons moving from the right end closer to the peak.
Yet this is not the only way to let nucleons switch to the higher binding energy state than the initial one. We can "resettle" very light elements (like hydrogen) and let nucleons move to the peak from the left. That would be fusion.
Heavy nucleons emerge in the stars. Here the gravitational energy is high enough to let the nucleons "unite" into whatever nuclei they like. Stars usually are formed from the very light elements and the nucleons inside, again, tend to get to the states with lower energies, and form more "medium-number" nuclei. The energy difference powers stars and we see the light emission, high temperatures and all other fun effects.
However, sometimes the temperatures in the stars are so high, that nucleons form the very heavy nuclei from the medium-number nuclei. even though there is no immediate "energy" benefit.
These heavy elements then disseminate everywhere with the death of the star. This stored star energy can then be released in the fission bomb.
Best Answer
I worked on a program two decades ago where we were to determine if nations "XYZ" were building nuclear weapons based on intelligence "ABC." Before I did this I took a DOE course that amounted to intermediate level (210) nuclear weapons. A lot of data on this is classified, and one needs CWDI Q-clearance to know this stuff. A lot of effort has gone into measuring just these physical parameters. After the 1963 nuclear test treaty restricted tests to underground (a really good thing) these tests used detectors and cables that measured these things. The bomb under ground would register data on a detector that is vaporized and the data would travel up transmission lines ahead to the destruction. A lot of this data collected is highly classified.
One can get a rule of thumb understanding of things. The temperature in the immediate environment of a nuclear bomb is about $5\times 10^7$K. Using Wein's law $\nu_{max}~\simeq~(2.9/hc)kT$ for the frequency at the black body peak for this temperature $\nu~=~3.3\times 10^{18}$Hz and equivalently $\lambda~=~9\times 10^{-11}m$. We can also calculate the energy $E~=~h\nu$ that is $12$KeV. This in the X-ray range of energy. If you were to place a photon detector in space to measure a nuclear burst this would be about where the peak of the EM spectrum would be.
These photons are are secondaries after interacting with the materials of the bomb. The initial nuclear induced photons are at higher energy. The nuclear process does not primarily generate photons, which are generated by QED interactions. However, the motion of fission and fusion products induced by the nuclear interaction produces photons as these ions scatter off of each other by their electrostatic potentials. These photons are in the $100$KeV to $1$ MeV range of energy.
Neutrons are produced, and in the case of fusion they constitute $18$MeV of energy produced per fusion $D~+~T~\rightarrow~{}_2^4He~+~n$, which in turn produce photons as secondaries when they interact with matter. There is the neutron bomb that is a $T-T$ nuclear bomb meant to produce lots of neutrons. These have a magnetic moment that they interact with matter, and these are damaging to biological molecules. The neutron bomb is then largely an anti-personnel weapon and fashioned into a mini-hydrogen bomb.
As a comment in general, it will be interesting to see if our species finally gets out of the obsession over these before we end up using them in a global war.