The heterogeneous nature of the navigation surface suggests adaptation capabilities in vehicle motion control to overcome the effects of the wheel-terrain interaction. In such scenario, this paper presents an integral adaptive control framework built upon a Nonlinear Moving Horizon Estimator and a Nonlinear Model Predictive Control scheme, under which the objective is to on-line estimate states and model parameters of a robot motion model while autonomously navigating in off-road terrain conditions. With an adjustable model, the controller is made adaptive against terrain changes while tracking prescribed trajectories. The system is composed by two coupled subsystems to represent the vehicle motion and tire slip dynamics. The combined control-estimation strategy works under the Real-Time Iteration scheme to attain reliable computational activity for high-speed tire dynamics (e.g., slip). Trials in a simulation and real test environment with a compact mini-loader Cat® 262C, as those found in the mining industry, showed that the approach is able to efficiently estimate states and model parameters without exceeding constraints. The analysis of computational efficiency in various hardware configurations is also provided, exhibiting that the rapid optimization involved in the proposed controller is possible for high-speed dynamics.