The magnetization dynamics of Co nanoislands on Cu(111) is studied on the atomic scale by means of the Landau-Lifshitz-Gilbert equation. The exchange and anisotropy constants of the spin Hamiltonian are computed from first principles. We focus on hysteresis loops and magnetic switching in dependence on temperature, island size, and strength of an external magnetic field. The magnetic switching of nanoislands whose magnetization is reversed on the sub-nanosecond time scale is found consistent with the Stoner-Wohlfarth theory. We separate the superparamagnetic from the ferromagnetic regime and provide evidence that nanodomains can exist at least on a sub-picosecond time scale.