In this paper, we study band-to-band and intersubband characteristics of GaN/AlN heterostructures (planar and nanowires) structurally designed to absorb in the short-wavelength infrared region, particularly at 1.55 microns. We compare the effect of doping the GaN sections with Si and Ge, and we discuss the variation of free-carrier screening with the doping density and well/nanodisk size. We observe that nanowire heterostructures consistently present longer photoluminescence decay times than their planar counterparts, which supports the existence of an in-plane piezoelectric field associated to the shear component of the strain tensor, leading to lateral electron-hole separation. We report intersubband absorption covering 1.45 microns to 1.75 microns using Ge-doped quantum wells, with comparable performance to well-studied Si-doped planar heterostructures. We also report comparable intersubband absorption in Si- and Ge-doped nanowire heterostructures indicating that the choice of dopant is not an intrinsic barrier for observing intersubband phenomena. In addition, we calculate the spectral shift of the intersubband absorption due to many body effects as a function of the doping concentration.