In a CuCl nanocrystal embedded in a NaCl single crystal, we observed a time response of exciton and biexciton luminescence intensities under strong pulsed light excitation by using the up-conversion method at 2 and 77 K. It was found that the rise of the biexciton luminescence is delayed compared to that of the exciton luminescence at 77 K. This is assumed to be because the biexciton is not directly created from electron-hole $(e\ensuremath{-}h)$ pairs but through exciton formation. On the basis of the cascade relaxation model, the formation time of an $e\ensuremath{-}h$ pair to an exciton is estimated to be 1.6 ps. We also observed a time response of luminescence intensities of the bound exciton, free biexciton, and bound biexciton at 2 K. These time profiles are interpreted by the cascade relaxation process. The formation time of the $e\ensuremath{-}h$ pair is estimated to be 1.3 ps, and the trapping lifetimes of the exciton to a surface impurity center and of the $e\ensuremath{-}h$ pair to the bound exciton are estimated to be 1.4 and 1.8 ps, respectively.