The upper critical field $H_{c2}$ of polycrystalline samples of $Ln$O$_{0.5}$F$_{0.5}$BiS$_{2}$ ($Ln$ = La, Nd) at ambient pressure (tetragonal structure) and high pressure (HP) (monoclinic structure) have been investigated via electrical resistivity measurements at various magnetic fields up to 8.5 T. The $H_{c2}$($T$) curves for all the samples show an uncharacteristic concave upward curvature at temperatures below $T_c$, which cannot be described by the conventional one-band Werthamer-Helfand-Hohenberg theory. For the LaO$_{0.5}$F$_{0.5}$BiS$_{2}$ sample under HP, as temperature is decreased, the upper critical field $H_{onset}$, estimated from the onset of the superconducting transitions, increases slowly between 4.9 and 5.8 T compared with the slope of $H_{onset}$($T$) below 4.9 T and above 5.8 T. This anomalous behavior reveals a remarkable similarity in superconductivity between LaO$_{0.5}$F$_{0.5}$BiS$_{2}$ samples measured under HP and synthesized under HP, although the crystal structures of the two samples were reported to be different. The experimental results support the idea that local atomic environment, which can be tuned by applying external pressure and can be quenched to ambient pressure via high temperature-pressure annealing, is possibly more essential to the enhancement of $T_c$ for BiS$_2$-based superconductors than the structural phase transition. On the other hand, such anomalous behavior is very subtle in the case of NdO$_{0.5}$F$_{0.5}$BiS$_{2}$ under HP, suggesting that the anisotropy of the upper critical field in the $ab$-plane and the possible lattice deformation induced by external pressure is weak. This explains why the pressure-induced enhancement of $T_c$ for NdO$_{0.5}$F$_{0.5}$BiS$_{2}$ is not as large as that for LaO$_{0.5}$F$_{0.5}$BiS$_{2}$.
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