how long does it take to fall into despair?

A few weeks to a few months probably. Assuming you start from a well-adjusted place. First, there is a shock which rocks the foundations of your worldview. Usually a person stays in shock for a few days... just numbness. Then they go through the various stages of grief, their self defence mechanisms kicking in. Nothing makes sense and that will make them angry and sad in alternating fashion. Then comes acceptance, which people assume will lead to healing. But if the world fundamentally stops making sense to the person, then the acceptance is an acceptance of life in a broken world. A few months pass by without a glimmer of hope, and that's when despair sets in.