The formation mechanisms of microbumps and nanojets on films composed of single and double Cu/Ag layers deposited on a glass substrate and irradiated by a single 60 fs laser pulse are investigated experimentally and in atomistic simulations. The composition of the laser-modified bilayers is probed with the energy dispersive X-ray spectroscopy and used as a marker for processes responsible for the modification of the film morphology. For the bilayer with the top Ag layer facing the laser, the increase in fluence is found to result in a sequential appearance of a Ag microbump, the exposure of the Cu underlayer by removal of the Ag layer, a Cu microbump, and a frozen nanojet. The Cu on Ag bilayer exhibits a partial spallation of the top Cu film, followed by the generation of surface structures that mainly consist of Ag at higher fluences. The experimental observations are explained with atomistic simulations, which reveal that the stronger electron-phonon coupling of Cu results in the confinement of the deposited laser energy in the top Cu layer in the Cu on Ag case and channelling of the energy from the top Ag layer to the underlying Cu layer in the Ag on Cu case. This difference in the energy (re)distribution directly translates into differences in the morphology of the laser-modified bilayers. In all systems, the generation of microbumps and nanojets occurs in the molten state. It is driven by the dynamic relaxation of the laser-induced stresses and, at higher fluences, the release of vapor at the interface with the substrate. The resistance of the colder periphery of the laser spot to the ejection of spalled layers as well as the rapid solidification of the transient molten structures are largely defining the final shapes of the surface structures.