In this work, we consider vortex lattices in rotating Bose-Einstein condensates composed of two species of bosons having different masses. Previously (Barnett et al 2008 New J. Phys. 10 043030), it was claimed that the vortices of the two species form bound pairs and the two vortex lattices lock. Remarkably, the two condensates and the external drive all rotate at different speeds owing to the disparity of the masses of the constituent bosons. In this paper, we study the system by solving the full two-component Gross-Pitaevskii equations numerically. Using this approach, we verify the stability of the putative locked state that is found to exist within a disc centered on the axis of rotation and that depends on the mass ratio of the two bosons. We also derive a refined estimate for the locking radius tailored to the experimentally relevant case of a harmonic trap and show that this agrees with the numerical results. Finally, we analyze in detail the rotation rates of the different components in the locked and unlocked regimes.