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We present the first 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(Pb+Pb→Pb+J/ψ+Pb), to NLO in the conventional collinear factorisation approach. 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(Pb+Pb→Pb+J/ψ+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 in this conventional approach: 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. We will also present an outlook for this observable within a tamed collinear factorisation, where we supplement the conventional approach with a programme of low x resummation and implementation of a crucial low Q subtraction. It will be shown how this opens up the possibility to constrain and ultimately determine the low x and low scale gluon PDF in the proton and heavy nuclei.