Non-terrestrial networks based on unmanned aerial vehicles (UAVs) are considered for integration with 6G mobile networks to provide ubiquitous communication through heterogeneous networks. UAVs offer benefits such as flexible deployment, mobility control, and strong line-of-sight links. In this paper, orthogonal multiple access (OMA) and nonorthogonal multiple access (NOMA) are combined to form a hybrid OMA-NOMA cognitive radio (CR) assisted cooperative UAV system. The selection of either the OMA or NOMA mode is carried out such that the best link quality is offered to the user equipments (UEs) compared to the individual modes. In this system, the base station uses the NOMA mode for the transmission to the UAVs which act as decode-and-forward relays each serving a cluster of UEs. The CR paradigm is used for the NOMA mode of the hybrid system to reduce the interference caused by the UAVs to adjacent clusters of UEs. Analytical expressions of the outage probability and sum rate of the OMA, NOMA, and hybrid OMA-NOMA systems are derived. Numerical results are provided to reveal the performance advantages of the proposed hybrid OMA-NOMA CR-assisted cooperative UAV system over Nakagami-m fading. Further, design options and tradeoffs between the OMA, NOMA, and hybrid OMA-NOMA systems are illustrated. © 2024 IEEE.