Ferromagnet/nonmagnet (FM/NM) bilayer-based spin Hall nano-oscillators (SHNOs)—a sub-class of spintronic oscillator devices—have promising potential toward realizing low-power physical reservoir computing systems because of their inherent nonlinearity and miniature form factor. However, most of the studies on SHNOs indicate that an external biasing magnetic field is necessary for their operation, creating a bottleneck for their practical implementation in designing small and compact RC hardware. In this report, using micromagnetic simulation, we demonstrate biasing field-free operation of a FM/NM bilayer-based SHNO by exploiting the magnetic anisotropy. Our results reveal that the magnetic anisotropy in the FM layer provides active control over the DC tunability of auto-oscillation frequency and the threshold value of current needed for sustained auto-oscillations. We show that the increase in uniaxial anisotropy substantially modifies the spatial profile of auto-oscillation and eventually leads to the reduction in the threshold current for auto-oscillation, which could be utilized to design low-power computing hardware using SHNO devices.