In this paper, we consider a non-orthogonal multiple access (NOMA)-enabled satellite-aerial-terrestrial network, where a batch of unmanned-aerial-vehicles (UAVS) act as decode-and-forward (DF) relays to simultaneously serve ground user equipments (UEs). The UAVs employ joint-transmission coordinated multi-point (JT-CoMP) to cooperatively provide coverage to a cluster of UEs utilizing the same resource block (RB), in a low signal-to-interference-plus-noise-ratio (SINR) setting. Our main objective is to increase the quality-of-service (QoS) of the UEs by maximizing the system sum-rate, which is accomplished by presenting a relay selection and a power allocation scheme, under limited available transmission power at the satellite and UAVs, QoS of UEs, and decoding order of UEs constraints. We first present an optimal UAV relays selection scheme which selects a group of UAVs satisfying the rate and decoding order constraints, and then sequentially solve the power allocation optimization problem for the two stages of transmission using Lagrange multipliers method and Karush-Kuhn-Tucker (KKT) conditions. Simulation results prove the effectiveness of our proposed system in successfully increasing the sum-rate of the network compared to baseline schemes, hence amplifying spectral efficiency.