Satellite communication is becoming more and more significant in present and future communication systems for its unique advantages of strong survivability and seamless coverage, which can offset the defects of terrestrial communication such as being restricted by terrain and little coverage. Moreover, unmanned aerial vehicle (UAV)-assisted communication is considered as a promising development direction due to its flexibility and expansibility in the integrated satellite-terrestrial networks (ISTN). In addition, to overcome the spectrum shortage and low spectrum utilization problems, cognitive radio and nonorthogonal multiple access (NOMA) has been widely utilized to enhance spectrum efficiency, which is considered as the key technologies of the next generation communication. In this regard, our paper investigates the performance of cognitive satellite-UAV-terrestrial networks with NOMA scheme and multiple primary users. Specifically, the exact expressions of outage probability (OP) and ergodic capacity for both primary network and secondary network are derived. To gain further views, the asymptotic expression of OP and diversity orders for the two networks are provided. Finally, through numerical simulations, the correctness of the theoretical derivations is verified and the influences of critical variables on system indexes are also analyzed.