Nanowires (NWs) with embedded zero-dimensional (0D) quantum dots (QDs) have interesting fundamental properties attractive for a variety of applications. The properties of such embedded QDs can be controlled by 0D quantum confinement and also via strain engineering in axial or radial heterostructures of the nanowire system. We evaluate the electronic structure of QDs, which are formed in the Ga(N, As) shell of the GaAs/Ga(N, As) core-shell NWs due to alloy fluctuations. It is found that the principal quantization axis of the studied QDs is primarily oriented along the NW axis, based on the performed polarizationresolved magneto-photoluminescence measurements. We also show that the QDs exhibit a large spectrally dependent variation of the valence band character, which changes from pure heavy-hole states for the lowenergy QD emitters to the mixed light-hole heavy-hole states for the QDs emitting at high energies. We ascribe these changes to combined effects of the uniaxial strain caused by the lattice mismatch between the GaAs core and the Ga(N, As) shell, and the local strain/lattice distortion within the short-range fluctuations in the N content. The obtained results underline the importance of the local strain for valence band engineering in hybrid NW structures with embedded QDs.