6G mobile networks are foreseen to integrate non-terrestrial networks (NTNs), such as unmanned aerial vehicle (UAV) networks, with conventional terrestrial networks to improve global coverage. UAVs offer flexible deployment and strong line-of-sight propagation links when placed at favorable altitudes. In this paper, we propose a hybrid amplify-and-forward (AF) decode-and-foward (DF) relaying based non-orthogonal multiple access (NOMA) UAV system. In this system, the base station uses NOMA to simultaneously communicate on the downlink with a number of hybrid AF-DF UAV relays which improves spectral efficiency. The UAVs select the AF or DF relaying mode such that the highest signal-to-interference-plus-noise ratio at the user equipments is obtained. Hence, either the low signal processing complexity in AF relaying mode or the potentially lower power dissipation in DF relaying mode is utilized. Analytical expressions of the outage probability and sum rate of the hybrid AF-DF relaying based NOMA UAV system are derived. Numerical results are provided revealing the performance gains offered by the proposed system in the presence of Nakagami-m fading and the system's benefits over AF and DF based systems.