Time-dependent specific heat experiments on the molecular nanomagnet Fe8 and the isotopic enriched analogue 57Fe8 are presented. The inclusion of the 57Fe nuclear spins leads to a huge enhancement of the specific heat below 1 K, ascribed to a strong increase in the spin-lattice relaxation rate Gamma arising from incoherent, nuclear-spin-mediated magnetic quantum tunneling in the ground-doublet. Since Gamma is found comparable to the expected tunneling rate, the latter process has to be inelastic. A model for the coupling of the tunneling levels to the lattice is presented. Under transverse field, a crossover from nuclear-spin-mediated to phonon-induced tunneling is observed.