Nanowire antennas embedding a single quantum dot (QD) have recently emerged as versatile platforms to realize bright sources of quantum light. In this theoretical work, we show that the thermally driven, low-frequency vibrations of the nanowire have a major impact on the QD light emission spectrum. Even at liquid helium temperatures, these prevent the emission of indistinguishable photons. To overcome this intrinsic limitation, we propose three designs that restore photon indistinguishability thanks to a specific engineering of the mechanical properties of the nanowire. We anticipate that such a mechanical optimization will also play a key role in the development of other high-performance light-matter interfaces based on nanostructures.