This paper presents a novel CFAR compressed detection approach for human detection using IR-UWB radar. The associated Xampling scheme operates way below the Nyquist limit and is designed to minimize the sensing matrix coherence, without increasing the implementation complexity. The proposed signal processing architecture aims to detect both moving and stationary people in the framework of heavy-cluttered use cases, such as smart factory indoor environments. In order to address this challenge, we rely on standard radar signal processing, including MTI filtering, noise whitening, and Doppler focusing, but also introduce two new algorithms for joint sparse reconstruction and CFAR detection, in fast-time and range-Doppler domains, respectively. We propose a specific detection statistic, which is proven to be appropriate for both algorithms, its distribution being identified and then validated by standard goodness-of-fit tests. Moreover, it enables reducing the CFAR scheme complexity, since the associated detection threshold is invariant to the noise power, thus making unnecessary its estimation. The proposed approach is finally validated using both simulated and experimentally measured data in an Industry 4.0 indoor environment, for several canonical scenarios. The effectiveness of our CFAR compressed detection algorithms for human detection is thus fully demonstrated, and their performance is assessed and compared to that obtained by signal processing at the Nyquist sampling rate.