In this paper, we propose a mixture gamma distribution based analytical framework for NOMA wireless systems over composite fading channels. We analyze the outage probability (OP), delay-limited throughput (TP) and effective capacity (EC) in uplink NOMA with imperfect successive interference cancellation (SIC) due to the presence of residual hardware impairments and delay constraints. A mixture gamma distribution is used to approximate the probability density functions of fading channels. Based on this, we obtain closed-form expressions in terms of Meijer-G functions for the OP, the TP and the EC. We also perform asymptotic analysis of these metrics to characterize system behaviors at the high signal-to-noise ratio regime. Moreover, upper-bounds for the EC is derived. Efficacy of NOMA over orthogonal multiple access is analytically examined. Unlike the existing works, our analytical expressions hold for NOMA systems with an arbitrary number of users per cluster over a wide range of channel models, including lognormal-Nakagami-m, KG, η-μ, Nakagami-q (Hoyt), κ-μ, Nakagami-n (Rician), Nakagami-m, and Rayleigh fading channels. This unified analysis facilitates evaluations of impacts of the residual interference, the power allocation among users, the delay quality-of-service exponent as well as the shadowing and small-scale fading parameters on the performance metrics. Simulation results are provided to validate theoretical analysis.