By truncating the range of van der Waals forces in our molecular dynamics model we reduce the lengthscale and timescale gap between the outer (wedge) and inner (precursor) regions in droplet spreading simulations. This results in a molecular model which combines atomic-scale resolution with the ability to capture large-scale behaviour as manifested by the Tanner spreading law. Our results show that Tanner's law can be recovered, even if van der Waals effects and the resulting precursor film are limited to distances of the order of three atomic diameters from the substrate. In other words, removal of the singularity is not necessary up to a few atomic diameters from the contact line. The very good quantitative agreement with theory and experiments suggests that the original precursor theory of de Gennes can be generalized to precursors of molecular thickness in which flow is not characterized by the continuum (Stokes) model. Gravity current simulations are also in excellent agreement with the theory of Huppert and recent experimental results showing lubrication scalings at small capillary numbers.