Nonlinear robust model predictive control (RMPC) provides a promising solution to the problem of automatic emergency maneuvering and it is capable of handling multiple, possibly conflicting objectives of robustness and performance. Even though RMPC gives a sub-optimal solution, the critical challenge in its real-time implementation is the high computational demand. This paper presents a real-time capable, robust tube MPC-based framework for steering control during emergency obstacle avoidance maneuvers. The novelty of this framework lies in the robust integration of path planning and path-following tasks of autonomous vehicles. A simulation study showcases the robust performance improvements of the proposed strategy over a non-robust MPC in different extreme maneuvering scenarios.