The lattice dynamics of oxygen-doped ${\mathrm{Nd}}_{2}\mathrm{Ni}{\mathrm{O}}_{4+\ensuremath{\delta}}$ is explored by a combination of single-crystal neutron spectroscopy, ab initio molecular dynamics (AIMD), and harmonic phonon calculations using density functional theory. The inelastic response computed from AIMD finds better correspondence with the experimental inelastic neutron spectra than the harmonic phonon calculations. The comparison of experimental and simulated data indicates that the presence of extra oxygen in interstitial sites leads to marked differences in the dynamics of the system compared to the stoichiometric parent compound. In particular, a nondispersive scattering signal was detected below 15 meV in all the experimental spectra in the $(ab)$ plane. Its origin has been rationalized as an important incoherent scattering from Nd atoms, which emerges from in-plane atomic self-motions in the ${\mathrm{Nd}}_{2}{\mathrm{O}}_{2+\ensuremath{\delta}}$ rock-salt layers. The presence of additional interstitial oxygen atoms both dampens the long-range lattice dynamics as well as enhances oxygen and neodymium anharmonic displacements. The experimental observation of coherent and incoherent responses allows assessing the effect of interstitials on the vibrations of the whole crystal as well as on the dynamics of each atom independently.