The second-order (v(2)) and third-order (v(3)) Fourier coefficients describing the azimuthal anisotropy of prompt and nonprompt (from b-hadron decays) J/ψ, as well as prompt ψ(2S) mesons are measured in lead-lead collisions at a center-of-mass energy per nucleon pair of √s(NN) = 5.02TeV. The analysis uses a data set corresponding to an integrated luminosity of 1.61 nb(-1) recorded with the CMS detector. The J/ψ and ψ(2S) mesons are reconstructed using their dimuon decay channel. The v(2) and v(3) coefficients are extracted using the scalar product method and studied as functions of meson transverse momentum and collision centrality. The measured v(2) values for prompt J/ψ mesons are found to be larger than those for nonprompt J/ψ mesons. The prompt J/ψ v(2) values at high pT are found to be underpredicted by a model incorporating only parton energy loss effects in a quark-gluon plasma medium. Prompt and nonprompt J/ψ meson v(3) and prompt ψ(2S) v(2) and v(3) values are also reported for the first time, providing new information about heavy quark interactions in the hot and dense medium created in heavy ion collisions.