The equilibration of electronic carriers in metals after excitation by an ultrashort laser pulse provides an important class of nonequilibrium phenomena in metals and allows measuring the effective electron-phonon coupling parameter. Since the observed decay of the electronic distribution is governed by the interplay of both electron-electron and electron-phonon scattering, the interpretation of experimental data must rely on models that ideally should be easy to handle, yet accurate. In this paper, an extended rate-equation model is proposed that explicitly includes nonthermal electronic carriers while at the same time incorporating data from first-principles calculations of the electron-phonon coupling via Eliashberg-Migdal theory. The model is verified against experimental data for thin Pb films grown on Si(111). Improved agreement between theory and experiment at short times (<0.3ps) due to nonthermal electron contributions is found. Moreover, the rate equations allow for widely different coupling strength to different phonon subsystems. Consequently, an indirect, electron-mediated energy transfer between strongly and weakly coupled groups of phonons can be observed in the simulations that leads to a retarded equilibration of the subsystems only after several picoseconds.