The answer is almost no time at all, a few collisions only, a tiny fraction of a second. The system is nonlinear, as are all real systems. This also means that they are irreversible, contrary to conventional assumptions. Well, the billiard ball model, like most models, is a simplified treatment. It ignores gravity for instance.
Atoms are subject to it in the same way as are planets.
This phenomena is known as sensitivity to initial conditions, or the Butterfly Effect. This behaviour is responsible for what we call Chaos, a term that has a technical meaning but is roughly equivalent to the common notion of future randomness, except that the states in which the system may be found are often restricted and known in total - we just don't know which one it will be in at any future time. Each molecule is acted upon by every object in the universe - an infinite sum of terms is thus necessary for an exact model. Well, we have four factors at play here. We can do this because even a small system contains vast numbers of particles.