Heterogeneous nucleation of water vapour on insoluble nuclei is a phenomenon that can induce atmospheric water and ice cloud formation. However, modelling of the phenomenon is hampered by the fact that the predictive capability of the classical heterogeneous nucleation theory is rather poor. A reliable theoretical description of the influence of different types of water-insoluble nuclei in triggering the water condensation or ice deposition would help to decrease uncertainty in large-scale model simulations. In this paper we extend a recently formulated adsorption theory of heterogeneous nucleation to be applicable to highly curved surfaces, and test the theory against laboratory data for water vapour nucleation on silica, titanium dioxide and silver oxide nanoparticles. We show that unlike the classical heterogeneous nucleation theory, the new theory is able to quantitatively predict the experimental results.