We consider the problem of nonlinear oscillatory convection in a horizontal mushy layer rotating about a vertical axis. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the stable and unstable oscillatory solutions of the weakly nonlinear problem by using perturbation and stability analyses. It was found that depending on the values of the parameters, supercritical simple travelling modes of convection in the form of hexagons, squares, rectangles or rolls can become stable and preferred, provided the value of the rotation parameter $\tau$ is not too small and is below some value, which can depend on the other parameter values. Each supercritical form of the oscillatory convection becomes subcritical as $\tau$ increases beyond some value, and each subcritical form of the oscillatory convection is unstable. In contrast to the non-rotating case, qualitative properties of the left-travelling modes of convection are different from those of the right-travelling modes, and such qualitative difference is found to be due to the interactions between the local solid fraction and the Coriolis term in the momentum-Darcy equation.