Semiconductor nanowires offer great potential for realizing broadband photodetectors that are compatible with silicon technology. However, the spectral range of such detectors has so far been limited to selected regions in the ultraviolet, visible and near infrared. Here, we report on broadband nanowire heterostructure array photodetectors exhibiting a photoresponse from the visible to long-wavelength infrared. In particular, the infrared response from 3-20 um is enabled by normal incidence excitation of intersubband transitions in low-bandgap InAsP quantum discs synthesized axially within InP nanowires. The optical characteristics are explained by the excitation of the longitudinal component of optical modes in the photonic crystal formed by the nanostructured portion of the detectors, combined with a non-symmetric potential profile of the discs resulting from synthesis. Our results provide a generalizable insight into how broadband nanowire photodetectors may be designed, and how engineered nanowire heterostructures open up new fascinating opportunities for optoelectronics.