The ${B}^{\ifmmode\pm\else\textpm\fi{}}$ meson production asymmetry in $pp$ collisions is measured using ${B}^{+}\ensuremath{\rightarrow}{\overline{D}}^{0}{\ensuremath{\pi}}^{+}$ decays. The data were recorded by the LHCb experiment during Run 1 of the LHC at center-of-mass energies of $\sqrt{s}=7$ and 8 TeV. The production asymmetries, integrated over transverse momenta in the range $2<{p}_{\mathrm{T}}<30\text{ }\text{ }\mathrm{GeV}/c$, and rapidities in the range $2.1<y<4.5$ are measured to be ${\mathcal{A}}_{\mathrm{prod}}({B}^{+},\sqrt{s}=7\text{ }\text{ }\mathrm{TeV})=\phantom{\rule{0ex}{0ex}}(\ensuremath{-}0.41\ifmmode\pm\else\textpm\fi{}0.49\ifmmode\pm\else\textpm\fi{}0.10)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, ${\mathcal{A}}_{\mathrm{prod}}({B}^{+},\sqrt{s}=8\text{ }\text{ }\mathrm{TeV})=(\ensuremath{-}0.53\ifmmode\pm\else\textpm\fi{}0.31\ifmmode\pm\else\textpm\fi{}0.10)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, where the first uncertainties are statistical and the second are systematic. These production asymmetries are used to correct the raw asymmetries of ${B}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{+}$ decays, thus allowing a measurement of the $CP$ asymmetry, ${\mathcal{A}}_{CP}({B}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{+})=(0.09\ifmmode\pm\else\textpm\fi{}0.27\ifmmode\pm\else\textpm\fi{}0.07)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$.