Single dopant nanowire transistors can be viewed as the ultimate miniaturization of nano electronic devices. In this work, we theoretically investigate the influence of the electron-phonon coupling on their transport properties using a non-equilibrium Green's function approach in the self-consistent Born approximation. For an impurity located at the center of the wire we find that, at room temperature, acoustic phonons broaden the impurity level so that the bistability predicted in the ballistic regime is suppressed. Optical phonons are found to have a beneficial impact on carrier transport via a phonon-assisted tunneling effect. We discuss the position and temperature dependence of these effects, showing that such systems might be very promising for engineering of ultimate devices.