The cross-sections of exclusive (coherent) photoproduction J/ψ and ψ(2S) mesons in ultra-peripheral PbPb collisions at a nucleon-nucleon centre-of-mass energy of 5.02TeV are measured using a data sample corresponding to an integrated luminosity of 228 ± 10 µb^(-1), collected by the LHCb experiment in 2018. The differential cross-sections are measured separately as a function of transverse momentum and rapidity in the nucleus-nucleus centre-of-mass frame for J/ψ and ψ(2S) mesons. The integrated cross-sections are measured to be (σ(J/ψ))^(coh) = 5.965 ± 0.059 ± 0.232 ± 0.262mb and (σ(ψ(2S))^(coh) = 0.923 ± 0.086 ± 0.028 ± 0.040mb, where the first listed uncertainty is statistical, the second systematic and the third due to the luminosity determination. The cross-section ratio is measured to be (σ(ψ(2S))^(coh)/(σ(J/ψ))^(coh) = 0.155 ± 0.014 ± 0.003, where the first uncertainty is statistical and the second is systematic. These results are compatible with theoretical predictions.