There is no such thing as miniaturized big bang.. The big bang itself is something that happens on nuclear level.. Not some kind of bomb explosion.. :]
That is not entirely true. Before and very soon after the Big Bang, there was no such thing as an atom, or nuclear force (strong or weak). It is believed that at such extreme energy, all of the forces were equally strong and can be expressed with the same equation (which is still being looked for in Grand Unification Theory). It did not happen on a nuclear level at all. However, you were right that it was not similar to a bomb's explosion.
By the way, miniaturized versions of the Big Bang
DO exist, they're called singularities, and they're found at the center of black holes. Inside of a singularity, the density is infinite, and so it is thought to be infinitely hot, just like the pre-Big Bang singularity.
This is taken from Stephen Hawking's A Brief History of Time:
"At the big bang itself the universe is thought to have had zero size, and so to have been infinitely hot. But as the
universe expanded, the temperature of the radiation decreased. One second after the big bang, it would have fallen
to about ten thousand million degrees. This is about a thousand times the temperature at the center of the sun, but
temperatures as high as this are reached in H-bomb explosions. At this time the universe would have contained
mostly photons, electrons, and neutrinos (extremely light particles that are affected only by the weak force and
gravity) and their antiparticles, together with some protons and neutrons."
We will understand much more with a working theory of quantum gravity. I think we'll see that theory in our lifetimes.