We explore the effect of the shell thickness on the time response of CdS/CdSe/CdS spherical quantum wells (SQWs) nanoscintillators under X-ray excitation. We first compare the spectral and timing properties under low and intense optical excitation, which allows us to identify the complex temporal and spectral response of the highly excited species. We find that a defect-induced delayed luminescence appears at large sizes. Under pulsed X-ray excitation, an analysis of the scintillation decay time reveals that multiexcitons are generated, similarly to the intense optical excitation and that the shell thickness does not change the fraction of fast component to a large extent. We performed a two-step simulation of the energy relaxation in the SQWs which reveals that large-size SQWs favor a very high number of excitations per particle, which, however, is counterbalanced by increased Auger quenching, rendering large SQWs less effective regarding the timing performance.