For unmanned aerial systems (UAS) to be successfully deployed and integrated within the national airspace, it is imperative that they possess the capability to effectively complete their missions without compromising the safety of other aircraft, as well as persons and property on the ground. This necessity creates a natural requirement for UAS that can respond to uncertain environmental conditions and emergent failures in real-time, with robustness and resilience close enough to those of manned systems. We introduce a system that meets this requirement with the design of a real-time onboard system health management (SHM) capability to continuously monitor sensors, software, and hardware components. This system can detect and diagnose failures and violations of safety or performance rules during the flight of a UAS. Our approach to SHM is three-pronged, providing: (1) real-time monitoring of sensor and software signals; (2) signal analysis, preprocessing, and advanced on-the-fly temporal and Bayesian probabilistic fault diagnosis; and (3) an unobtrusive, lightweight, read-only, low-power realization using Field Programmable Gate Arrays (FPGAs) that avoids overburdening limited computing resources or costly re-certification of flight software. We call this approach rt-R2U2, a name derived from its requirements. Our implementation provides a novel approach of combining modular building blocks, integrating responsive runtime monitoring of temporal logic system safety requirements with model-based diagnosis and Bayesian network-based probabilistic analysis. We demonstrate this approach using actual flight data from the NASA Swift UAS.