Rare-earth ($R$) kagome-lattice magnets offer an excellent platform to discover the novel magnetic phase as well as quantum phase transition tuned by nonthermal control parameters, while the experimental realizations remain largely unexplored. Here, we report the discovery of magnetic field ($B$) -induced anisotropic quantum phase transition in a distorted kagome antiferromagnet ${\mathrm{Nd}}_{3}{\mathrm{BWO}}_{9}$ with ${T}_{\mathrm{N}}\ensuremath{\sim}0.32\phantom{\rule{0.28em}{0ex}}\mathrm{K}$. The isothermal magnetizations at 0.05 K exhibit the spin-flop-like metamagnetic crossover behaviors with different fractional magnetization anomalies for $B$ perpendicular ($B||c$ axis) and parallel ($B||{a}^{*}$ axis) to the kagome plane, respectively. In combination with the thermodynamic measurements, the field-temperature ($B\text{\ensuremath{-}}T$) phase diagrams for both field directions are constructed and reveal the existence of several field-induced magnetic states. Along the $c$ axis, a proximate quantum bicritical point is observed near the metamagnetic crossover, which separates the low-field antiferromagnetic (AFM) phase and the intermediate AFM phase; while, for $B||{a}^{*}$, another intermediate magnetic phase (IAFM2) appears between the low-field AFM phase and the intermediate AFM (IAFM1) phase, giving rise to a tetracritical point. These results support the anisotropic field-induced metamagnetic quantum criticalities in ${\mathrm{Nd}}_{3}{\mathrm{BWO}}_{9}$, making it a rare kagome antiferromagnet to investigate the quantum multicriticality driven by spin frustration.