Radioactivity though spontaneous all the atoms don't disintegrate at a time
The basic concept of radioactive decay kinetics is that every single nucleus has the same probability of decaying at any single given time, regardless of the physical-chemical surroundings (it does not know anything about the other atoms). This of course means that they do not disintegrate all at once, because the probability of all of them "deciding to decay" at the exact same time is infinitesimally small.
Spontaneous breakdown has no external cause. To a close approximation, breakdown products do not cause/trigger/contribute to subsequent decay events. The other-than-spontaneous breakdown possibilities are slender because nuclei are small and well-separated inside their electron shells.
Neutron emission (in fissile materials) is the notable exception, because neutral particles don't interact with electron charge. The usual neutron energies after a uranium-235 decay are too high to hit many targets, but with some purification of isotopes and a bunch of moderator material that slows neutrons down (a 'core', usually graphite), you can get a chain reaction (analogous to kindling a fire).
In the Oklo region, isotopic abundances indicate that chain reactions were kindled naturally in the distant past, with water as the moderator.