On the basis of an ab initio theory of metallic magnetism in layered materials, we investigate the onset of magnetic order in thin (2-8 layers) fcc-Fe films on and embedded in Cu(100) substrates. In particular, we find an oscillatory dependence of the Curie temperatures on embedding depth, in excellent agreement with experimental data. The thermally induced spin fluctuations are treated within a mean-field disordered local moment picture and give rise to layer-dependent "local exchange splittings" in the electronic structure even in the paramagnetic phase. These features determine the magnetic intralayer and interlayer interactions which are strongly influenced by the presence and extent of the Cu cap.