Cost-effective mid-wave infrared (MWIR) optoelectronic devices are of utmost importance to a plethora of applications such as night vision, thermal sensing, autonomous vehicles, free-space communication, and spectroscopy. To this end, leveraging the ubiquitous silicon-based processing has emerged as a powerful strategy that can be accomplished through the use of group IV germanium-tin (GeSn) alloys. Indeed, due to their compatibility with silicon and their tunable bandgap energy covering the entire MWIR range, GeSn semiconductors are frontrunner platforms for compact and scalable MWIR technologies. However, the GeSn large lattice parameter has been a major hurdle limiting the quality of GeSn epitaxy on silicon wafers. Herein, it is shown that sub-20 nm Ge nanowires (NWs) provide effective compliant substrates to grow Ge$_{1-x}$Sn$_{x}$ alloys with a composition uniformity over several micrometers with a very limited build-up of the compressive strain. Ge/Ge$_{1-x}$Sn$_{x}$ core/shell NWs with Sn content spanning the 6 to 18 at.$\%$ range are demonstrated and integrated in photoconductive devices exhibiting a high signal-to-noise ratio at room temperature and a tunable cutoff wavelength covering the 2.0 $\mu$m to 3.9 $\mu$m range. Additionally, the processed NW-based detectors were used in uncooled imagers enabling the acquisition of high-quality images under both broadband and laser illuminations without a lock-in technique.