The contributions to the free energy of a nematic wetting layer as a function of its thickness l are analyzed. The longest-range contribution is due to distortion of the nematic director across the film, resulting from different preferred molecular orientations at the two interfaces bounding the film. Van der Waals forces as well as the decaying tails of the interfacial order-parameter profiles yield contributions to the free energy of successively shorter range. These effects lead to crossovers between different scaling régimes for variation of the mean wetting-layer thickness with temperature. Experimental implications of the results are described.