In this work, we apply the space–time Galerkin reduced basis (ST–GRB) method to a reduced fluid–structure interaction model, for the numerical simulation of hæmodynamics in arteries. In essence, ST–GRB extends the classical reduced basis (RB) method, exploiting a data–driven low–dimensional linear encoding of the temporal dynamics to further cut the computational costs. The current investigation brings forth two key enhancements, compared to previous works on the topic. On the one side, we model blood flow through the Navier–Stokes equations, hence accounting for convection. In this regard, we implement a hyper–reduction scheme, based on approximate space–time reduced affine decompositions, to deal with nonlinearities effectively. On the other side, we move beyond the constraint of modelling blood vessels as rigid structures, acknowledging the importance of elasticity for the accurate simulation of complex blood flow patterns. To limit computational complexity, we adopt the Coupled Momentum model, incorporating the effect of wall compliance in the fluid's equations through a generalized Robin boundary condition. In particular, we propose an efficient strategy for handling the spatio–temporal projection of the structural displacement, which ultimately configures as a by–product. The performances of ST–GRB are assessed in three different numerical experiments. The results confirm that the proposed approach can outperform the classical RB method, yielding precise approximations of high–fidelity solutions at more convenient costs. However, the computational gains of ST–GRB vanish if the number of retained temporal modes is too large, which occurs either when complex dynamics arise or if very precise solutions are sought.