Quantum dots inserted in semiconducting nanowires are an interesting platform for the fabrication of single photon devices. To fully understand the physical properties of these objects we need to correlate the optical, transport and structural properties on the same nanostructure. In this work, we study the spectral tunability of the emission of a single quantum dot in a GaN nanowire by applying external bias. The nanowires are dispersed and contacted on electron beam transparent Si3N4 membranes, so that transmission electron microscopy observations, photocurrent and micro-photoluminescence measurements under bias can be performed on the same specimen. The emission from a single dot blue or red shifts when the external electric field compensates or enhances the internal electric field generated by the spontaneous and piezoelectric polarization. A detailed study of two nanowire specimens emitting at 327.5 nm and 307.5 nm shows spectral shifts at rates of 20 and 12 meV/V, respectively. Theoretical calculations facilitated by the modelling of the exact heterostructure provide a good description of the experimental observations. When the bias-induced band bending is strong enough to favor tunneling of the electron in the dot towards the stem or the cap, the spectral shift saturates and additional transitions associated to charged excitons can be observed.