The nuclear modification factors of J/psi and psi(2S) mesons are measured in PbPb collisions at a centre-of-mass energy per nucleon pair of root S-NN = 5.02 TeV. The analysis is based on PbPb and pp data samples collected by CMS at the LHC in 2015, corresponding to integrated luminosities of 464 mu b(-1) and 28 pb(-1), respectively. The measurements are performed in the dimuon rapidity range of vertical bar y vertical bar < 2.4 as a function of centrality, rapidity, and transverse momentum (p(T)) from p(T) = 3 GeV/c in the most forward region and up to 50 GeV/c. Both prompt and nonprompt (coming from b hadron decays) J/psi mesons are observed to be increasingly suppressed with centrality, with a magnitude similar to the one observed at root S-NN = 2.76 TeV for the two J/psi meson components. No dependence on rapidity is observed for either prompt or nonprompt J/psi mesons. An indication of a lower prompt J/psi meson suppression at p(T) > 25 GeV/c is seen with respect to that observed at intermediate p(T). The prompt psi(2S) meson yield is found to be more suppressed than that of the prompt J/psi mesons in the entire p(T) range.