Background: The LHCb Collaboration has studied a number of kinematic correlations between $B$-hadron pairs through their subsequent decays to $J/\psi$ pairs in $p+p$ collisions at 7 and 8 TeV for four minimum values of the $J/\psi p_T$. Purpose: In this work, these measurements are compared to calculations of $b\overline{b}$ pairs and their hadronization and inclusive decays to $J/\psi J/\psi$ are compared to the same observables. Potential cold matter effects on the $b\overline{b}$ pair observables are discussed to determine which are most likely to provide insights about the system and why. Methods: The calculations, employing the exclusive HVQMNR code, assume the same intrinsic $k_T$-broadening and fragmentation as in [R. Vogt, Phys. Rev. C 98, 034907 (2018)]. The pair distributions presented by LHCb are calculated in this approach, both for the parent $b\overline{b}$ and the $J/\psi J/\psi$ pairs produced in their decays. The sensitivity of the results to the intrinsic $k_T$ broadening is shown. The theoretical uncertainties due to the $b$ quark mass and scale variations on both the initial $b\overline{b}$ pairs and the resulting $J/\psi$ pairs are also shown, as is the dependence of the results on the rapidity range of the measurement. Possible effects due to the presence of the nucleus are studied by increasing the size of the $k_T$ broadening and modifying the fragmentation function. Results: Good agreement with the LHCb data is found for all observables. The parent $b\overline{b}$ distributions are more sensitive to the $k_T$ broadening than are the final-state $J/\psi$ pairs. Conclusions: Next-to-leading order calculations with $k_T$ broadening, as in [R. Vogt, Phys. Rev. C 98, 034907 (2018)], can describe all correlated observables. Multiple measurements of correlated observables are sensitive to different nuclear effects which can help distinguish between them.