The space-air-ground integrated network (SAGIN) greatly improves coverage and reliability for millimeter-wave (mmWave) communication in high-speed railway (HSR) scenarios. However, a significant challenge arises in the transmission scheduling due to the rapid changes in channel state, link selection for train mobile relays (MRs), and order of the flow scheduling. To tackle this challenge, we introduce an optimization problem focused on maximizing the weighted sum completed flows that satisfy the quality of service (QoS) requirements for HSR mmWave communication in SAGIN. To facilitate the simultaneous scheduling of flows by base station-MR (BS-MR), satellite-airship-MR, and satellite-MR links, we propose a link selection algorithm, which can help each flow choose a suitable set of links in every frame and determine whether the BS networks need the assistance of the satellite and airship. Furthermore, taking into account the priority and occupied time slots (TSs) resource of different flows, we propose a multi-link weighted flow scheduling (MWFS) algorithm. This algorithm not only prioritizes scheduling high-priority flows but also aims to maximize the weighted sum completed flows for MRs. Our simulation results confirm that the proposed algorithm significantly increases the weighted sum completed flows and the total transmitted bits. Additionally, the proposed algorithm can achieve the optimal flow transmission in different link switching periods and enhance the scheduling of high-priority flows compared to other algorithms.