In space–air–ground integrated network (SAGIN), the high mobility of Low Earth Orbit (LEO) satellites results in short communication windows, posing significant challenges to aggregation efficiency and latency control in traditional centralized federated learning (FL) frameworks. To address these challenges, this paper proposes a low-latency and high-performance FL framework tailored for SAGIN. The system consists of ground users, base stations, UAVs, and LEO satellites, and is capable of efficient communication and model aggregation under dynamic connectivity conditions. The convergence behavior of the proposed framework is theoretically analyzed under standard assumptions. To further enhance the learning performance, an optimization problem is formulated that minimizes the weighted sum of the user association weight deviation and training delay. The formulation also incorporates satellite communication delay constraints to reflect the limited contact durations between satellites and ground nodes. The problem is decomposed into user association and wireless resource allocation subproblems, which are jointly solved using a primal dual iterative algorithm. Simulation results based on the non-independent and identically distributed (no-IID) distribution of the CIFAR-10 and VisDrone2019-DET dataset demonstrate that the proposed method achieves faster convergence speed and lower training latency than baseline algorithms under different network scales, validating the effectiveness and practicality of the proposed framework.