We consider sparsity-driven joint localization of stationary and moving targets inside enclosed structures using a reduced set of spatial-fast time-slow time observations in ultra-wideband (UWB) pulsed radar platforms. We exploit the compact temporal support of the UWB signal to suppress the front wall clutter through time gating. The resulting enhancement in the signal-to-clutter ratio enables application of compressive sensing (CS) for scene reconstruction. We establish an appropriate signal model that permits formulation of linear modeling with sensing matrices so as to achieve efficient CS-based localization of stationary and moving targets in the downrange-crossrange-velocity space. We demonstrate the effectiveness of the proposed scheme using real data collected in a laboratory environment.