  In this paper, we calculate the total decay widths for the $W^+$-boson
decays, $W^+ \to B_c+b+\bar{s}+X$ and $W^+ \to B^*_c+b+\bar{s}+X$, up to
next-to-leading order (NLO) accuracy within the framework of the
nonrelativistic QCD theory. Both the fixed-order and the fragmentation
approaches are adopted to do the calculation. Differential decay widths
$d\Gamma/dz$ and $d\Gamma/ds_1$ are also given. We find that the NLO
corrections are significant in those two $W^+$ decay channels. Our numerical
results show that at the LHC, there are about $7.03\times 10^4$ $B_c$-meson
events and $5.10\times 10^4$ $B^*_c$-meson events to be produced via the
$W^+$-boson decays per operation year.
