We compute the $O\left({\alpha }_s^2\right)$ corrections to the exclusive channels $e^{+}{e}^{-}\to J/\psi +{\chi }_{cJ}$ ($J=0,1,2$) at $\sqrt{s}=10.58$ GeV within the nonrelativistic QCD (NRQCD) factorization framework. The validity of NRQCD factorization at $O\left({\alpha }_s^2\right)$ has been confirmed for these double-charmonium exclusive production processes. We analyze the impact of the $O\left({\alpha }_s^2\right)$ corrections on the polarized and unpolarized cross sections, as well as the $J/\psi$ angular distributions, which largely reduce the renormalization scale dependence but increase the $O\left({\alpha }_s\right)$ NRQCD predictions to some extent for ${\chi }_{c0,1}$. With high numerical accuracy, our predictions for $\sigma (J/\psi +{\chi }_{c1,2})$ through $O\left({\alpha }_s^2\right)$ are compatible with the upper limit of the Belle measurement. Although the theoretical prediction for $\sigma (J/\psi +{\chi }_{c0})$ is consistent with both Belle and BaBar measurements within uncertainties, there still exists serious tension between the predicted and the measured profiles for the $J/\psi$ angular distribution. Regarding the bright observation prospect of the $e^{+}{e}^{-}\to J/\psi +{\chi }_{c1,2}$ channels in Belle 2 experiment, it is interesting to compare the future measurements with our NRQCD predictions. The more accurate measurement of $e^{+}{e}^{-}\to J/\psi +{\chi }_0$ at Belle 2 will also help to clarify the long-standing puzzle of $J/\psi$ angular distribution.