Despite the widespread deployment of quadrotor unmanned aerial vehicles (UAVs), ensuring their flight stability under asymmetric environmental disturbances, such as concurrent wind and rain, remains a significant challenge. To address the trajectory tracking problem under these severe conditions, this paper proposes a Composite Continuous Rapid Nonsingular Adaptive Terminal Sliding Mode (CCRNATSM) control strategy. First, a composite dynamic model is developed, integrating wind aerodynamics with rain impact characteristics to accurately simulate realistic flight environments. A High-Order Sliding Mode Observer (HOSMO) is then employed for the real-time, accurate estimation of these lumped disturbances. Subsequently, this observer is integrated with an adaptive control law to ensure rapid and precise system stabilization. Comparative simulations conducted under strong disturbance conditions demonstrate that the proposed method exhibits superior performance over existing strategies, reducing roll angle deviation by 75% and shortening the recovery time to 1.5 s. Ultimately, this control strategy significantly enhances the robustness and safety of quadrotor UAVs operating in harsh, asymmetric environments.