We study the nonlinear development of an oscillatory convective instability in a two-dimensional mushy layer during solidification of a binary mixture. We adopt the near-eutectic limit, making the problem analytically tractable using standard perturbation techniques. We consider also a distinguished limit of large Stefan number, which allows a destabilization of the system to the oscillatory mode of convection. We provide a comprehensive investigation of the nonlinear stability of travelling and standing waves, including the relative stability between the two wave patterns. In addition to mapping out the location of different stable oscillatory patterns in the available parameter space, we give the detailed spatio--temporal structure of the corresponding thermal, flow and solid fraction fields within the mushy layer, as well as the local bulk composition in the resulting eutectic solid. The relevance of our results to recent laboratory experiments is also discussed.