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In this paper, we propose a novel adaptive decoding mechanism (ADM) for the unmanned aerial vehicle (UAV)-enabled uplink (UL) non-orthogonal multiple access (NOMA) communications. Specifically, considering a harsh UAV environment, where ground-to-ground links are regularly unavailable, the proposed ADM overcomes the challenging problem of conventional UL-NOMA systems whose performance is sensitive to the transmitter's statistical channel state information and the receiver's decoding order. To evaluate the performance of the ADM, we derive closed-form expressions for the system outage probability (OP) and system throughput. In the performance analysis section, we provide novel expressions for practical air-to-ground and ground-to-air channels, while taking into account the practical implementation of imperfect successive interference cancellation (SIC) in UL-NOMA. Moreover, the obtained expression can be adopted to characterize the OP of various systems under a Mixture of Gamma (MG) distribution-based fading channels. Next, we propose a sub-optimal Gradient Descent-based algorithm to obtain the power allocation coefficients that result in maximum throughput with respect to each location on UAV's trajectory. To determine the significance of the proposed ADM in nonstationary environments, we consider the ground users and the UAV to move according to the Random Waypoint Mobility (RWM) and Reference Point Group Mobility (RPGM) models, respectively. Accurate formulas for the distance distributions are also provided. Numerical solutions demonstrate that the ADM-enhanced NOMA not only outperforms Orthogonal Multiple Access (OMA), but also improves the performance of UAV-enabled UL-NOMA even in mobile environments.