In this paper, the uplink outage performance of double-reconfigurable intelligent surface (RIS)-assisted non-orthogonal and orthogonal multiple access (NOMA/OMA) systems is investigated. Specifically, we consider a two-user network operating over Nakagami-m fading channels, where each RIS enhances the signal of its corresponding user while introducing random interference to the other user. To accurately characterize the system's outage behavior, a Gamma approximation via the moment matching technique is employed to derive tractable closed-form expressions for the outage probability (OP) of both users, accounting for coherent and non-coherent signal components. The analytical framework captures the effects of key system parameters, including the number of RIS elements and fading severity. Furthermore, Monte Carlo simulations validate the analytical results, demonstrating that (i ) double-RIS NOMA achieves significant power-efficiency gains over OMA, (ii) a pronounced cliff effect appears in the strong user's OP beyond a critical allocation threshold, and (iii) optimal element allocation between the two user-dedicated RISs is highly sensitive to path-loss conditions. These results provide valuable insights for the design and optimization of multi-RIS-assisted NOMA/OMA systems.