This paper investigates the performance of non-orthogonal multiple access (NOMA) based hybrid satellite-unmanned aerial vehicle (UAV) systems, where a low Earth orbit (LEO) satellite communicates with the ground users via a decode and forward (DF) UAV relay. We investigate a two NOMA users system, where a far user (FU) and a near user (NU) are served by the UAV which is located at a certain height above the origin of the coverage circle. The channel between satellite and UAV is assumed to follow a Shadowed-Rician fading and the channels between UAV and users are assumed to follow a Nakagami-m fading. New closed-form expressions of the outage probabilities for the two users and the system are derived. Different from other work in literature, we take into consideration different parameters affecting the total link budget. Additionally, we propose an algorithm for minimizing the system outage probability. The mathematical analysis is verified by extensive representative Monte-Carlo (MC) simulations. Finally, simulations are provided to demonstrate the impact of important parameters on the considered system as well as the superiority of the NOMA scheme the over reference scheme.