Silicon carbide based composites are candidates for structural components and fuel claddings in nuclear power plants. In the frame of accident tolerant fuel research, the effective through-thickness thermal conductivity of SiC/SiC prototype claddings is investigated. This property, which is both material and geometry dependent, has been measured with a custom-made radial heat flow apparatus. Conductivities ranging between 0.5 and $\sim4$~W$\cdot$m$^{-1}\cdot$K$^{-1}$ have been measured, well below the literature values of 8 to 15~W$\cdot$m$^{-1}\cdot$K$^{-1}$. This significant difference is due to several factors. Firstly, with the method used here, the components of the conductivity, parallel, and perpendicular to the fibre weave, are fully separated. Secondly, the complex multilayered architectures of the tubes are detrimental, acting as barriers against heat transport. Indeed, in order to be water- and gas-tight, the tubes have to include dense ceramic or metallic layers. The poor adhesion between these and SiC/SiC results in a severely reduced heat transfer. Moreover, the matrix of the composite shows a heterogeneously distributed porosity --~ about 20 to 45\%. This leads to an inhomogeneous thermal conductivity, along both the length and the circumference of the cladding. Although the total porosity should be lessened in future development steps, it currently contributes to the transport of heat through Mie forward scattering of infrared radiation, an effect known as radiation thermal conductivity. In parallel to these activities, a specific component of SiC/SiC, the pyrolytic carbon interphase, has been investigated using analytical electron microscopy. This work showed that this layer, linking the fibres to the matrix, becomes partially amorphous after irradiation. In-situ ion irradiations evidenced radiation-induced dimensional changes of the said layer, the impact of which would mostly be on the mechanical properties of the composite. Using continuous medium modelling, the effect of a partial amorphisation of the PyC is estimated to a thermal conductivity loss of 20 to 25\%. In their current state, SiC/SiC clads do not have a thermal conductivity high enough for them to used in power plants. Indeed, a thermal conductivity locally as low as 1~W$\cdot$m$^{-1}\cdot$K$^{-1}$ results in fuel temperatures high enough for \ce{UO2} to reach its melting point.