The production of J/ψ mesons in heavy-ion collisions at the Large Hadron Collider is believed to be dominated by the recombination of charm and anticharm quarks in a hot QCD medium. However, measurements of the elliptic flow (v_{2}) of J/ψ mesons in these reactions are not well described by existing calculations of J/ψ recombination for transverse momenta p_{T}≳4 GeV. We revisit these calculations in two main aspects. Employing the resonance recombination model, we implement distribution functions of charm quarks transported through the quark-gluon plasma using state-of-the-art Langevin simulations and account for the space-momentum correlations of the diffusing charm and anticharm quarks in a hydrodynamically expanding fireball. This extends the relevance of the recombination processes to substantially larger momenta than before. We also revisit the suppression of primordially produced J/ψ's by propagating them through the same hydrodynamic medium, leading to a marked increase of their v_{2} over previous estimates. Combining these developments into a calculation of the p_{T}-dependent nuclear modification factor and v_{2} of inclusive J/ψ production in semicentral Pb-Pb collisions at the LHC, we find a good description of the experimental results by the ALICE Collaboration. Our results thus resolve the abovementioned v_{2} puzzle and imply the relevance of recombination processes for p_{T}'s of up to ∼8 GeV.