This work examines the influence of quantum dots embedded in semiconductor nanowires with lattice parameters different from those of the surrounding nanowire material on the nanowire band gap. Using the found analytical formulas for the elastic fields of cylindrical, hemispherical, and conical inclusions simulating quantum dots and located along the nanowire symmetry axis, the maps of the elastic dilations are depicted and the regions of their extremes near the nanowire surface are identified. Calculations are performed within the framework of the isotropic linear theory of elasticity. For GaN nanowires containing axisymmetric quantum dots of varying shapes and compositions, the deformation potentials and corresponding changes in the band gap in the nanowire regions of elastic dilation extremes are calculated. The dependence of local change in the band gap in GaN nanowires on the lattice mismatch parameter between the quantum dots and NWs are presented. It is shown that the semiconductor nanowire band structure depends on the quantum dot shape, material, and size, and the band gap of GaN nanowires can locally vary by approximately 10 % of its tabulated value. The response of the band gap of a semiconductor nanowire to the elastic field of quantum dot embedded in this wire allows one to nanowire band gap engineering by varying the parameters of the quantum dot.