Molecular photoionization delays are often analyzed by assuming that nuclei remain fixed during the ionization process since they move much more slowly than electrons. However, recent high-energy resolution and multicoincidence experiments have shown that nuclear motion can have a significant and visible effect on the measured ionization delays on the attosecond time scale. To analyze this behavior, we have chosen the simplest of all molecules, H2+, and performed nearly exact calculations of streaking and RABBIT (reconstruction of attosecond beatings by interferences in two-photon transitions) spectra by solving the time-dependent Schrödinger equation in full dimensionality, and retrieved the corresponding photoionization delays. We show that, when two-center effects are at play, nuclear motion is responsible for a substantial increase of the photoionization delays, in particular of the so-called continuum-continuum delays. The magnitude of such an increase is comparable to the absolute values of the measured delays. Published by the American Physical Society 2024