We apply a single-order-parameter Cahn-Hilliard theory to deduce properties of the fluid-crystal interface from nucleation experiments: The two Cahn-Hilliard parameters (the free energy scale and the coefficient of the square-gradient term) are chosen so that the experimentally determined interfacial free energy of nuclei is recovered. The theory is then used to predict the thickness and free energy of the equilibrium planar interface, and other quantities such as the Tolman length and characteristic thickness, which describe the curvature dependence of the interfacial free energy. The accuracy of the method is demonstrated on systems (Lennard-Jones and ice-water) for which these properties are known. Experimental data available for five stoichiometric oxide glasses are then analyzed. The reduced interfacial free energy (Turnbull's alpha) and the interface thickness, we obtained, cover the alpha=0.28-0.51 and the d=0.8-1.6 nm ranges. For oxide glasses we find that alpha scales with n(-1/3), where n is the number of molecules per formula unit. In agreement with computer simulations, the Tolman length is strongly size dependent, while far weaker though still perceptible temperature dependence is observed for the characteristic interface thickness used in Granasy's phenomenological diffuse interface theory. In some cases bulk crystal properties prevail at the center of nuclei, while in other systems the nuclei are ramified, and the local properties significantly deviate from those of the macroscopic crystal. The accuracy of these results rests on a hypothesized temperature independence of the Cahn-Hilliard parameters, an assumption whose validity remains to be seen at large undercoolings.