We investigate the effect of hydrodynamic interactions (HIs) on the crystal nucleation of hard-sphere colloids for varying supersaturations. We use molecular dynamics and stochastic rotation dynamics techniques to account for the HIs. For high supersaturation values, we perform brute force simulations and compute the nucleation rate, obtaining good agreement with previous studies where HIs were neglected. In order to access low supersaturation values, we use a seeding approach method and perform simulations with and without HIs. We compute the nucleation rates for the two cases and surprisingly find good agreement between them. The nucleation rate in both cases follows the trend of the previous numerical results, thereby corroborating the discrepancy between experiments and simulations. Furthermore, we investigate the amount of fivefold symmetric clusters (FSCs) in a supersaturated fluid under different physical conditions, following the idea that FSCs compete against nucleation. To this end, we explore the role of the softness of the pair interactions, different solvent viscosities, and different sedimentation rates in simulations that include HIs. We do not find significant variations in the amount of FSCs, which might reflect the irrelevance of these three features on the nucleation process.