Although the capillary spreading of a drop on a dry substrate is well studied, understanding and describing the physical mechanisms that govern the dynamics remain challenging. Here we study the dynamics of spreading of partially wetting nanodroplets by combining molecular dynamics simulations and continuum phase field simulations. The phase field simulations account for all the relevant hydrodynamics, i.e., capillarity, inertia, and viscous stresses. By coordinated continuum and molecular dynamics simulations, the macroscopic model parameters are extracted. For a Lennard-Jones fluid spreading on a planar surface, the liquid slip at the solid substrate is found to be significant, in fact crucial for the motion of the contact line. Evaluation of the different contributions to the energy transfer shows that the liquid slip generates dissipation of the same order as the bulk viscous dissipation or the energy transfer to kinetic energy. We also study the dynamics of spreading on a substrate with a periodic nanostructure. Here it is found that a nanostructure with a length scale commensurate with molecular size completely inhibits the liquid slip. The dynamic spreading is thus about 30% slower on a nanostructured surface compared to one that is atomically smooth.