This paper studies the performance of the uplink Non-Orthogonal Multiple Access (NOMA) with Joint Maximum-Likelihood Detector (JML). We present a generalized upper bound of bit-error rate (BER) expression of adaptive M-ary phase-shift keying (M-PSK) over Rayleigh fading channels. Our studies are enriched through the use of a single transmitting antenna by each user and multiple receiving antennas at the base station (single-input-multiple-output (SIMO)). The derivation of the upper bound expression is obtained considering an arbitrary constellation size by performing Maximum Ratio Combining (MRC) of diversity paths. The extensive computer simulations validate the analysis and it is revealed that the JML outperforms the existing detecting algorithms significantly and achieves the full diversity order for each user.