Carbon nanotubes (CNTs) and boron nitride nanotubes (BNNTs) are one-dimensional materials with high thermal conductivity and similar crystal structures. Additionally, BNNTs feature higher thermal stability in air than CNTs. In this work, a single-walled carbon nanotube (SWCNT) film was used as a template to synthesize a BNNT coating by the chemical vapor deposition (CVD) method to form a coaxial heterostructure. Then, a contact-free steady-state infrared (IR) method was adopted to measure the in-plane sheet thermal conductance of the as-synthesized film. The heterostructured SWCNT-BNNT film demonstrates an enhanced sheet thermal conductance compared with the bare SWCNT film. The increase in sheet thermal conductance shows a reverse relationship with SWCNT film transparency. An enhancement of over 80% (from ∼3.6 to ∼6.4 μW·K-1·sq-1) is attained when the BNNT coating is applied to an SWCNT film with a transparency of 87%. This increase is achieved by BNNTs serving as an additional thermal conducting path. The relationship between the thermal conductance increase and transparency of the SWCNT film is studied by a structured modeling of the SWCNT film. We also discuss the effect of annealing on the thermal conductance of SWCNTs before BNNT growth. Along with the preservation of high electrical conductance, the enhanced thermal conductance of the heterostructured SWCNT-BNNT films makes them a promising building block for thermal and optoelectronic applications.