As energy storage devices are becoming more highly integrated, it is inevitable that heat accumulation will occur under high power working conditions. Finding efficient thermal management materials for cooling down electronic components is an urgent problem for energy storage devices. In this work, a thermally conductive film with tailorable macroproperties is fabricated by using a simple vacuum filtration method, using cellulose nanofibrils as the polymer substrate and assembly with aluminum nitride nanosheets. The interaction between the cellulose nanofibrils and aluminum nitride nanosheets is studied, and the electronic components made using the composite exhibit excellent thermal conductivity, thermal stability and mechanical flexibility. A high thermal conductivity is achieved along the film surface (up to 4.20 W/mK for 25 wt.% of aluminum nitride). This green material can effectively promote potential applications as lateral heat spreaders in flexible energy storage devices and the thermal conductivity may facilitate the applications in thermal management.