The understanding of heat transport in nonequilibrium thermodynamics is an important research frontier, which is crucial for implementing novel thermodynamic devices, such as heat engines and refrigerators. The convection, conduction, and radiation are the well-known basic ways to transfer thermal energy. Here, we demonstrate a different mechanism of phonon heat transport between two spatially separated nanomechanical resonators coupled by the cavity-enhanced long-range interactions. The single trajectory for thermalization and non-equilibrium dynamics is monitored in real-time. In the strong coupling regime, the instant heat flux spontaneously oscillates back and forth in the nonequilibrium steady states. The universal bound on the precision of nonequilibrium steady-state heat flux, i.e. the thermodynamic uncertainty relation, is verified in such a temperature gradient driven far-off equilibrium system. Our results give more insight into the heat transfer with nanomechanical oscillators, and provide a playground for testing fundamental theories in non-equilibrium thermodynamics.