We consider a two-patch Susceptible-Infected-Recovered epidemic model that incorporates awareness-driven behavioural changes in both contact and mobility patterns. Individuals modify their behaviour in response to a perceived risk of infection, which is modelled through two awareness variables that depend either on the current or past disease prevalence in each patch. We qualitatively analyse the model through stability and bifurcation theory and derive threshold conditions that determine the existence and stability of the biologically relevant equilibria. We find that awareness-induced behavioural changes in contact and mobility can destabilise mixed equilibria – where the disease persists in one patch only – and contribute to the emergence of stable co-endemic states. When awareness depends on past epidemic values, the stability analysis shows that mixed equilibria may lose stability via Hopf bifurcations, depending on the sign of some awareness-related parameters. Finally, the impact of the behaviour-related parameters on the epidemic dynamics is investigated through numerical simulations.