Coaxial core-shell nanowire heterostructures can consist of different materials with varying lattice parameters, electronic band energetics, opposite doping types, and also of different crystal structures. Based on this high degree of freedom, core-shell nanowires open up a large variety of new concepts for applications on the nanoscale. Here, we demonstrate the controlled tuning of the GaN band gap within coaxial nanowire heterostructures by up to 240 meV towards higher, as well as towards lower energies. This is realized by the epitaxial overgrowth of GaN nanowires by (Al,Ga)N and (In,Ga)N shells inducing compressive and tensile strain in the GaN cores, respectively. Long-term stability tests are performed on GaN-(Al,Ga)N core-shell nanowires, revealing coherently strained crystals despite high strain fields of up to -3.4%. A reduction of the radiative recombination rate in GaN-(Al,Ga)N core-shell nanowires compared to pure GaN nanowires measured under optical excitation is discussed by means of strain-dependent dipole transition matrix elements. A study on the thermal dissociation of excitonic recombination is performed via temperature-dependent photoluminescence spectroscopy, indicating the passivation of nonradiative surface defect centers in the presence of an (Al,Ga)N shell. For the case of GaN-(In,Ga)N core-shell nanowires, complementary strain fields are measured within the core and the shell, which provides a new concept for band gap tuning in the visible spectral range. In addition, specific shapes of the (In,Ga)N top facets of the core-shell nanowires imply a new and simple method for evaluating the wurtzite crystal polarity of individual GaN nanowires.