A numerical model based on the diffuse interface approach is developed to simulate the phase separation of binary Upper Critical Solution Temperature (UCST) mixtures with critical and off-critical compositions in 2D geometry. The modeling is intended to simulate the response of such mixtures whilst cooled into the unstable region or to the metastable region. With an off-critical composition, the initial homogeneous mixture may separate via nucleation and growth mechanism, provided it is exposed to a strong enough composition perturbation. In this study, the numerical model is used to explore the response of off-critical composition mixtures to white noise and two different forms of nuclei: a circular drop, and a 2D wavelet. The significant differences in the phase separation processes taking place with critical solution via spinodal decomposition, and off-critical compositions via nucleation and growth, are demonstrated and discussed. While a white noise perturbation in concentration suffices to trigger separation in the unstable region, nuclei exceeding a critical size are required to initiate the separation in the metastable region.