Resource-Constrained Project Scheduling Problems (RCPSPs) are NP-complete, which makes it challenging to efficiently solve large instances and robustify solutions in the presence of uncertainty. To remedy this, we learn to efficiently mimic the solutions produced by Constraint Programming (CP) solver, using a Graph Neural Network (GNN) architecture designed to capture the structure of RCPSPs. Since the GNN solution may violate constraints, we ensure schedule feasibility at inference time by extracting the task ordering from the GNN schedule and post-processing it with the well-known Schedule Generation Scheme (SGS). We find that SIREN, the resulting algorithm, produces schedules that are of higher quality than those produced by the CP solver within the same computation time budget. The speed and solution quality of SIREN make it suitable as a component of an on-line scenario-based optimisation procedure for RCPSPs with stochastic durations. This leads to the SERENE system, which robustly selects, in real-time, the best next tasks to start in order to minimise the average makespan over the scenarios. Empirically, SERENE achieves better average makespan over different realisations of uncertainty than deterministic algorithms that continuously reschedule on the basis of either the worst, best or average task durations.