The carbon nanotubes' resilience to mechanical deformation is a potentially important feature for imparting tunable properties at the nanoscale. Using nonequilibrium molecular dynamics and empirical interatomic potentials, we examine the thermal conductivity variations with bending in the thermal transport regime where both ballistic and diffusive effects coexist. These simulations are enabled by the realistic atomic-scale descriptions of uniformly curved and buckled nanotube morphologies obtained by imposing objective boundary conditions. We uncover a contrasting behavior. At shorter lengths, the phonon propagation is affected significantly by the occurrence of localized structural buckling. As the nanotube length becomes comparable with the phonon mean free path, heat transport becomes insensitive to the buckling deformations. Our result settles the controversy around the differences between the current experimental and molecular-dynamics measurements of the thermal transport in bent nanotubes.