Ultrafast X-ray diffraction is used to quantify the transport of energy in laser-excited nanoscale gold-nickel (Au-Ni) bilayers. Electron transport and efficient electron-phonon coupling in Ni convert the laser-deposited energy in the conduction electrons within a few picoseconds into a strong non-equilibrium between hot Ni and cold Au phonons at the bilayer interface. Modeling of the subsequent equilibration dynamics within various two-temperature models confirms that for ultrathin Au films, the thermal transport is dominated by phonons instead of conduction electrons because of the weak electron-phonon coupling in Au.