Unsteady fluid flow around oscillating and rotating domains is a central challenge in computational mechanics, with applications spanning marine propellers, turbomachinery, and energy systems. High-fidelity numerical simulations of such flows remain computationally expensive due to moving boundaries, mesh distortion, and long-time integration requirements, limiting their use in iterative design and real-time prediction. To address these challenges, we present a graph neural network (GNN)-based surrogate framework for the spatio-temporal prediction of incompressible fluid flow around rotating rigid bodies, built upon our finite element-inspired hypergraph neural network (ϕ-GNN) architecture. The method operates directly on unstructured meshes by representing element–node connectivity as a hypergraph, enabling message passing that closely mirrors local stiffness assembly and information propagation in finite element formulations. The computational domain is partitioned into co-rotating and stationary sub-domains, separated by a dynamically adapting interface layer. Dedicated velocity and pressure reconstruction and re-projection schemes are introduced to maintain accuracy across mesh distortion and connectivity changes without explicit interpolation, while rotational equivariance is enforced through appropriate feature transformations. The proposed framework is validated on two benchmark problems: (i) two-dimensional flow around an oscillating wing and (ii) three-dimensional flow past a rotating cube. Results demonstrate stable long-horizon rollout predictions over hundreds to thousands of time steps, with accurate recovery of velocity, pressure, and integrated force coefficients compared to high-fidelity CFD solutions. Furthermore, incorporating sparse pressure sensor measurements enables a digital-twin formulation that bounds error growth and enforces accurate predictions over extended time horizons. These results highlight the potential of hypergraph-based GNN surrogates as efficient and scalable computational mechanics tools for modeling unsteady flows with oscillating and rotating structures, providing a foundation for future extensions to complex systems such as marine propellers and turbomachinery.