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We present the first next-to-leading-order (NLO) perturbative QCD (pQCD) study of rapidity-differential cross sections of coherent exclusive photoproduction of $J/ψ$ mesons in heavy-ion ultraperipheral collisions (UPCs) at the LHC, $dσ/dy(\text{Pb} + \text{Pb} \rightarrow \text{Pb} + J/ψ+ \text{Pb})$. For this, we account for the photon-nucleon NLO cross sections at the forward limit, the $t$ dependence using a standard nuclear form factor, and the photon fluxes of the colliding nuclei. Approximating the generalized parton distributions with their forward-limit parton distribution functions (PDFs), we quantify the NLO contributions in the cross sections, show that the real part of the amplitude and quark-PDF contributions must not be neglected, quantify the uncertainties arising from the scale-choice and PDFs, and compare our results with ALICE, CMS and LHCb $J/ψ$ photoproduction data in Pb+Pb UPCs, exclusive $J/ψ$ photoproduction data from HERA, and LHCb data in p+p. The scale dependence in $dσ/dy(\text{Pb} + \text{Pb} \rightarrow \text{Pb} + J/ψ+ \text{Pb})$ is significant, but we can find a scale-choice that reproduces the Pb+Pb UPC data both at 2.76 and 5.02 TeV collision energies. This process has traditionally been suggested to be a direct probe of nuclear gluon distributions. We show that the situation changes rather dramatically from LO to NLO: the NLO cross sections reflect the nuclear effects of both gluons and quarks in a complicated manner where the relative signs of the LO and NLO terms in the amplitude play a significant role.