The availability of orbit information with high precision and low latency is a key requirement for many Earth-observation missions, predominantly in the field of radio occultation. Traditionally, precise orbit determination solutions of low-Earth orbit (LEO) satellites are obtained offline on ground after downloading GNSS measurements and auxiliary spacecraft data to the processing center. The latency of this processing depends on the frequency of LEO downlink contacts and the availability of precise GNSS orbit and clock products required for the orbit determination process. These dependencies can be removed by computing the precise orbit determination solution on board the satellite using GNSS broadcast ephemerides. In this study, both real data and simulated measurements from a representative LEO satellite are processed in a flight-proven Kalman-filter algorithm. The paper studies the use of GPS, Galileo and BeiDou-3 for real-time orbit determination in different combinations with simulated measurements. Results show that use of dual-frequency observations and broadcast ephemerides of Galileo and BeiDou-3 leads to a significant reduction of 3D rms orbit errors compared to GPS-only processing. An onboard navigation accuracy of about one decimeter can be achieved without external augmentation data, which opens up new prospects for conducting relevant parts of the science data processing in future space missions directly on board a LEO satellite.