Novel liquid and charge transport behaviour can emerge in nanofluidics depending on the molecular-level interactions at aqueous interfaces, and provide new venues for water desalination and "blue'' energy conversion. Improving on our molecular-level picture of the interfacial structure of water and electrolyte solutions, provided by the so-called electric double layer (EDL), is key to further our understanding of osmotic transport in nanofluidics. Yet, a molecular-level picture of the EDL is to a large extent still lacking. Particularly, the role of the electronic structure has not been considered in the description of the diffuse layer. Here, we report enhanced sampling simulations based on ab initio molecular dynamics, aiming at unravelling the free energy of prototypical ions adsorbed at the aqueous graphene and hBN interfaces, and its consequences on nanofluidic osmotic transport. Specifically, we predicted the zeta potential, the diffusio-osmotic mobility and the diffusio-osmotic conductivity for a wide range of salt concentrations from the water and ion spatial distributions through an analytical framework based on Stokes equation and a modified Poisson-Boltzmann equation. Concentration-dependent scaling laws are observed, together with dramatic differences in osmotic transport between the two interfaces, including diffusio-osmotic flow and current reversal on hBN, but not on graphene. We could rationalize the results for the three osmotic responses with a simple model based on characteristic length scales for ion and water adsorption at the surface, which are found to be quite different on graphene and on hBN. Our work provides first principle insights into the structure and osmotic transport of aqueous electrolytes on two-dimensional materials and explores new pathways for efficient water desalination and osmotic energy conversion.