GaN/AlN nanowire heterostructures can display photoluminescence (PL) decay times on the order of microseconds that persist up to room temperature. Doping the GaN nanodisk insertions with Ge can reduce these PL decay times by two orders of magnitude. These phenomena are explained by the three-dimensional electric field distribution within the GaN nanodisks, which has an axial component in the range of a few MV/cm associated to the spontaneous and piezoelectric polarization, and a radial piezoelectric contribution associated to the shear components of the lattice strain. At low dopant concentrations, a large electron-hole separation in both the axial and radial directions is present. The relatively weak radial electric fields, which are about one order of magnitude smaller than the axial fields, are rapidly screened by doping. This bidirectional screening leads to a radial and axial centralization of the hole underneath the electron, and consequently, to large decreases in PL decay times, in addition to luminescence blue shifts.