This work presents a model predictive control (MPC) scheme to achieve three-dimensional (3D) tracking control and point stabilisation of an autonomous underwater vehicle (AUV) subject to environmental disturbances. The AUV is modelled as a coupled, nonlinear system. The control scheme is developed using a linear parameter-varying (LPV) formulation of the nonlinear model in velocity form to obtain an optimisation control problem with efficient online solvers and does not require model augmentation that can potentially increase computational efforts. The control strategy inherently provides offset-free control when tracking piece-wise constant reference signals, ensures feasibility for trajectories containing unreachable points and is relatively simple to implement since parameterisation of all equilibria is not required. A simple switching law is proposed for task switching between the 3D trajectory tracking and point stabilisation. The MPC is designed to ensure closed-loop stability of the vehicle in both motion control tasks via the imposition of terminal constraints. Through simulations of the coupled nonlinear Naminow-D AUV under ocean current and wave disturbances, the effectiveness of the control strategy in trajectory tracking and point stabilisation is demonstrated.