We introduce a novel framework of continuous-time ultra-wideband-inertial sensor fusion for online motion estimation. Quaternion-based cubic cumulative B-splines are exploited for parameterizing motion states continuously over time. Systematic derivations of analytic kinematic interpolations and spatial differentiations are further provided. Based thereon, a new sliding-window spline fitting scheme is established for asynchronous multi-sensor fusion and online calibration. We conduct a dedicated validation of the quaternion spline fitting method, and evaluate the proposed system, SFUISE (spline fusion-based ultra-wideband-inertial state estimation), in real-world scenarios using public data set and experiments. The proposed sensor fusion system is real-time capable and delivers superior performance over state-of-the-art discrete-time schemes. We release the source code and own experimental data at https://github.com/KIT-ISAS/SFUISE.