Counter-rotating circularly polarized, time-delayed pulses generate electron wave packets that interfere in momentum space and form spiral-shaped free-electron vortices. By numerically solving the time-dependent Schr"odinger equation beyond the Born-Oppenheimer approximation for one-photon single ionization of $H_2$, we show that these spirals rotate within the polarization plane as a function of the nuclear energy. A kinematical model, based on the instantaneous sharing of photon energy between the freed electron and the parent molecular ion, predicts this rotation and agrees quantitatively with the simulations. We further demonstrate that integrating over vibrational energies diminishes the visibility of the interference structure. Results from fixed-nuclei and atom-like reference calculations resemble single vibrational channels, whereas incoherent summation over multiple vibrational states leads to blurring and noticeable shifts of the apparent rotation. The joint-energy spectrum exhibits pronounced electron-nuclear correlation lines. In the case of co-rotating pulses, these lines are related to the interference rings in the momentum distribution. Conversely, for counter-rotating pulses, they originate from a dipolar modulation of the spiral pattern, caused by the nonisotropic molecular orbital.