Deep neural networks are increasingly being explored as surrogate models for nonlinear Model Predictive Control (MPC), motivated by their ability to approximate complex optimal control laws more quickly and at reduced online computational cost. However, the effect of approximate schemes on the robustness, structure, and closed-loop behaviour of tube-based MPC schemes has not yet been thoroughly analyzed. This work proposes an imitation-learning-based tube MPC framework for discrete-time nonlinear systems with additive disturbances, in which neural networks approximate nominal control. The control input is decomposed into a learned nominal component and a stabilizing linear feedback term. The robustness guarantees are preserved through modified invariant tube construction. The overall error dynamics are generated by a linearized closed-loop system with additive disturbances that capture model mismatch, external disturbances, and neural network approximation errors. The main contribution lies in the systematic construction of tightening sets and their analysis under the neural network's approximation error. Approximation errors are bounded probabilistically using tools from statistical learning. These are propagated through the error dynamics to construct robust positively invariant tubes that guarantee recursive feasibility and constraint satisfaction. The proposed methodology is developed within the ERC Consolidator Grant project DREAM-ON and targets applications in Structural Health Monitoring, mainly contributing to the project's control loop between a physical asset and its digital twin. Numerical validation on representative test cases is performed to verify the theoretical robustness guarantees.