In contrast to artificial metamaterials, natural two-dimensional (2D) hyperbolic materials are able to support higher electromagnetic confinement and stronger photonic density of states. This means natural 2D hyperbolic materials have fascinating potential for applications in thermal photonics. In this work, we investigate mechanisms of near-field thermal radiation (NFTR) in a ${\mathrm{T}}_{d}$--${\mathrm{WTe}}_{2}$ single layer at the nanoscale using the fluctuation-dissipation theorem and density-functional theory. The results show that NFTR of ${\mathrm{T}}_{d}$--${\mathrm{WTe}}_{2}$ can be 3 orders of magnitude larger than the blackbody limit. By using the photonic tunneling coefficient and the plasmon dispersion, we present a comprehensive study of plasmonic properties of ${\mathrm{T}}_{d}$--${\mathrm{WTe}}_{2}$. Moreover, our first-principles calculations predict that application of a certain amount of mechanical stress can trigger topological transition between the elliptic and the hyperbolic surface states by regulating the Fermi surface and interband excitation threshold. Lastly, we systematically exhibit the evolution of surface plasmon polaritons in ${\mathrm{T}}_{d}$--${\mathrm{WTe}}_{2}$ under mechanical stress and analyze the performance of using mechanical stress to modulate the corresponding NFTR. Our work explores the great potential of tunable radiative heat flux in ${\mathrm{T}}_{d}$--${\mathrm{WTe}}_{2}$, which is explained here through the peculiar nature of hyperbolic surface plasmons.