This study presents a comprehensive analysis of stability, bifurcation, and chaotic dynamics in the generalized Koren-Feingold cloud-rain system, incorporating time delay in rain formation. The model is analyzed analytically by determining the existence and stability of equilibrium points, proving the occurrence of the double Hopf bifurcation. Our results demonstrate that time delay plays a crucial role in dictating the dynamical transitions of the system. A small delay maintains system stability, whereas an increase beyond a critical threshold induces periodic oscillations through a Hopf bifurcation. Further increments lead to period-doubling bifurcations, chaotic attractors, and intermittent oscillatory regimes, demonstrating the formation of intricate meteorological phenomena. Numerical simulations reinforce these findings, illustrating that prolonged delays can lead to persistent cloud formation, irregular precipitation patterns, and chaotic fluctuations in the dynamics of rain. These insights deepen our understanding of cloud-rain interactions and provide valuable implications for improving weather prediction models, particularly in capturing extreme precipitation events influenced by atmospheric memory effects.