We have performed nuclear quadrupole resonance and nuclear magnetic resonance measurements on UCoAl with strong Ising-type anisotropy under $b$- and $c$-axes uniaxial pressure. In the $b$-axis uniaxial pressure ($P_{parallel b}$) measurement, we observed an increase in the metamagnetic transition field with increasing $P_{parallel b}$. In the $c$-axis uniaxial pressure ($P_{parallel c}$) measurement, on the other hand, we observed a ferromagnetic transition in zero magnetic field along the $c$-axis above $P_{parallel c}$ = 0.08 GPa. The anomaly of the nuclear spin-lattice relaxation rate divided by the temperature $left[ (T_1 T)^{-1} right]$ at $T$ = 20 K is suppressed by $P_{parallel b}$ and slightly enhanced by $P_{parallel c}$. The anisotropic uniaxial pressure response indicates that uniaxial pressure is a good parameter for tuning the Ising magnetism in UCoAl.
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