In this paper, we investigate a coordinated direct and relay transmission for uplink (CDRT-UL) of a cooperative non-orthogonal multiple access (NOMA) system, in which two users communicate with a base station (BS) with the aid of a decode-and-forward relay. By incorporating both relay links and non-negligible direct links from two users to BS, the CDRT-UL scheme forms an orthogonal structure at the BS receiver side, so that the BS receiver applies only linear combinations to recover the desired signals without abusing successive interference cancellation. Outage probability of the paired two users is derived in closed-form under independent but not necessarily identically distributed Nakagami-m fading channels. Asymptotic analysis in the high signal-to-noise ratio regime is also introduced that provides closed-form expressions, which indicates that both two users achieve the same asymptotic outage performance. It is also shown that an improved outage performance and the higher target rate can be achieved by the CDRT-UL scheme over those of the DF relay without direct links. The enhanced system performance achieved by the CDRT-UL scheme is corroborated by Monte Carlo simulations.