  Dibismuth dioxychalcogenides, Bi$_2$O$_2$Ch (Ch = S, Se, Te) are emerging
class of materials for next generation electronics and thermoelectrics with an
ultrahigh carrier mobility and excellent air stability. Among these,
Bi$_2$O$_2$S is fascinating because of stereochemically active 6$s^2$ lone pair
of Bi$^{3+}$ cation, heterogeneous bonding and high mass contrast of
constituent elements. In this work, we systematically investigate the effect of
hydrostatic pressure and its implications on lattice dynamics and phonon
transport properties of Bi$_2$O$_2$S by employing first principles calculations
along with the Boltzmann transport theory. The ambient $Pnmn$ phase exhibits a
low average lattice thermal conductivity ($\kappa_l$) of 1.71 W-m/K at 300 K.
We also find that Bi$_2$O$_2$S undergoes a structural phase transition from low
symmetry ($Pnmn$) to a high symmetry ($I4/mmm$) structure around 4 GPa due to
the Bi$^{3+}$ cation centering. Upon compression the lone pair activity of
Bi$^{3+}$ cation is suppressed, which increases $\kappa_l$ by nearly 3 times to
4.92 W-m/K at 5 GPa for $I4/mmm$ phase. The calculated phonon lifetimes and
Gr\"uneisen parameters show that anharmonicity reduces with increasing pressure
due to further suppression of lone pair, strengthening of intra and inter
molecular interactions, which raises the average room temperature $\kappa_l$ to
12.82 W-m/K at 20 GPa. Overall, the present study provides a comprehensive
understanding of hydrostatic pressure effects on stereochemical activity of the
Bi$^{3+}$ cation lone pair and its consequences on phonon transport properties
of Bi$_2$O$_2$S.
