As an important technique of enabling global access for 6G, low earth orbit satellite (LEO-Sat) communication is still facing the challenges of large path-loss, and severe Doppler effect. This paper studies the reliability performance of a downlink LEO-Sat communication system, where the orthogonal time frequency space (OTFS) scheme is employed to combat severe Doppler effect. Further, the unmanned aerial vehicle (UAV) based cooperative transmission is developed, in order to compensate for the large margin of path-loss caused by the long transmission distance. In particular, the closed-form expression for the outage probability of the OTFS based LEO-Sat transmission is derived, where the novel moment matching approach is used to closely approximate the PDF of a sum of shadowing Rician (SR) variables. Further, the condition of using UAV cooperation is obtained so that the positive reliability gain is guaranteed. Finally, both the theoretical analysis and simulation results show that, the OTFS scheme can significantly outperform the traditional OFDM scheme, while demanding to carefully address the trade-off between reliability improvement and implementation complexity determined by modulation size.