This paper presents a robust control strategy for trajectory-tracking control of Skid-Steer Mobile Manipulators (SSMMs) using a Robust Nonlinear Model Predictive Control (R-NMPC) approach that minimises trajectory-tracking errors while overcoming model uncertainties and terra-mechanical disturbances. The proposed strategy is aimed at counteracting the effects of disturbances caused by the slip phenomena through the wheel-terrain contact and bidirectional interactions propagated by mechanical coupling between the SSMM base and arm. These interactions are modelled using a coupled nonlinear dynamic framework that integrates bounded uncertainties for the mobile base and arm joints. The model is developed based on principles of full-body energy balance and link torques. Then, a centralized control architecture integrates a nominal NMPC (disturbance-free) and ancillary controller based on Active Disturbance-Rejection Control (ADRC) to strengthen control robustness, operating the full system dynamics as a single robotic body. While the NMPC strategy is responsible for the trajectory-tracking control task, the ADRC leverages an Extended State Observer (ESO) to quantify the impact of external disturbances. Then, the ADRC is devoted to compensating for external disturbances and uncertainties stemming from the model mismatch between the nominal representation and the actual system response. Simulation and field experiments conducted on an assembled Pioneer 3P-AT base and Katana 6M180 robotic arm under terrain constraints demonstrate the effectiveness of the proposed method. Compared to non-robust controllers, the R-NMPC approach significantly reduced trajectory-tracking errors by 79.5% for mobile bases and 42.3% for robot arms. These results highlight the potential to enhance robust performance and resource efficiency in complex navigation conditions.