The large data sample of the $B_c$ meson collected at the LHC experiment and the HL-LHC experiment provides us the opportunity to study the $B_c$ decays and the related physics. In this paper, we investigate the effect of $K^0-{\overline{K}}^0$ mixing on the branching ratios, $CP$ violations and $CPT$ violations in the $B_c^{\pm }\to B^{\pm }{K}_{S,L}^0$ decays. We find that some of the $B_c^{\pm }\to B^{\pm }{K}_{S,L}^0\to f_{B^{\pm }}{f}_{K_{S,L}^0}$ decay chains have large branching ratios, whose maximum value can exceed the order of ${10}^{-6}$, the minimum number of $B_c^{\pm }$ events times efficiency for observing the decays at three standard deviations ($3\sigma$) level is about ${10}^6$. We study the $CP$ asymmetries in the $B_c^{\pm }\to B^{\pm }{K}_{S,L}^0$ decays and find that the $CP$ asymmetries can exceed the order of ${10}^{-3}$, which are dominated by $K^0-{\overline{K}}^0$ mixing. We give the most promising processes to observe the $CP$ violations and the ranges of the numbers of $B_c^{\pm }$ events-times-efficiency needed to observe the $CP$ asymmetries at a significance of $3\sigma$ in these decays. We investigate the possibility to constraint the $CPT$ violation parameter $z$ in the $B_c^{\pm }\to B^{\pm }{K}_{S,L}^0\to f_{B^{\pm }}{f}_{K_{S,L}^0}$ decays and give the most promising processes to constraint the parameter $z$.