Gallium nitride (GaN) nanowires exist in a myriad of cross-sectional shapes. In this study, a series of classical molecular dynamics simulations is performed to investigate the strain-phononics-structure relationship in rectangular and triangular wurtzite-GaN nanowires. The thermal conductivity of the nanowires is linearly dependent on the uniaxial strain in both compressive and tensile regimes, and shows no significant dissimilitude for the same amount of strain exerted on the two types of nanowire. This is coherent with an analytical approach using the Boltzmann transport theory. However, the thermomechanical behaviour at the vertex regions shows palpable differences between the two subfamilies, relative to the non-vertex faceted regions, as the structural morphology is most disparate at the vertices. Furthermore, the degree of strain asymmetry is a strong determinant of the vibrational response and consequently thermal conductance.