A study of B and Bs meson decays into J/ψ η and J/ψ η′ final states is performed using a data set of proton-proton collisions at centre-of-mass energies of 7 and 8 TeV, collected by the LCHb experiment and corresponding to 3.0 fb−1 of integrated luminosity. The decay B0 → J/ψ η′ is observed for the first time. The following ratios of branching fractions are measured: $$ \begin{array}{c}\hfill \frac{\mathrm{\mathcal{B}}\left({\mathrm{B}}^0\to \mathrm{J}/{\uppsi \upeta}^{\prime}\right)}{\mathrm{\mathcal{B}}\left({\mathrm{B}}_{\mathrm{s}}^0\to\ \mathrm{J}/{\uppsi \upeta}^{\prime}\right)} = \left(2.28\pm 0.65\ \left(\mathrm{stat}\right)\pm 0.10\ \left(\mathrm{syst}\right)\pm 0.13\ \left(\mathrm{fs}/\mathrm{f}\mathrm{d}\right)\right)\times 1{0}^{-2},\hfill \\ {}\hfill \frac{\mathrm{\mathcal{B}}\left({\mathrm{B}}^0\to\ \mathrm{J}/\uppsi \upeta \right)}{\mathrm{\mathcal{B}}\left({\mathrm{B}}_{\mathrm{s}}^0\to\ \mathrm{J}/\uppsi \upeta \right)} = \left(1.85\pm 0.61\ \left(\mathrm{stat}\right)\pm 0.09\ \left(\mathrm{syst}\right)\pm 0.11\ \left(\mathrm{fs}/\mathrm{f}\mathrm{d}\right)\right)\times 1{0}^{-2},\hfill \end{array} $$ where the third uncertainty is related to the present knowledge of fs/fd, the ratio between the probabilities for a b quark to form a Bs or a B0 meson. The branching fraction ratios are used to determine the parameters of η − η′ meson mixing. In addition, the first evidence for the decay Bs → ψ(2S)η′ is reported, and the relative branching fraction is measured, $$ \frac{\mathrm{\mathcal{B}}\left({\mathrm{B}}_{\mathrm{s}}^0\to\ \uppsi \left(2\mathrm{S}\right){\upeta}^{\prime}\right)}{\mathrm{\mathcal{B}}\left({\mathrm{B}}_{\mathrm{s}}^0\to\ \mathrm{J}/{\uppsi \upeta}^{\prime}\right)} = \left(38.7\pm 9.0\ \left(\mathrm{stat}\right)\pm 1.3\ \left(\mathrm{syst}\right)\pm 0.9\left(\mathrm{\mathcal{B}}\right)\right)\times 1{0}^{-2}, $$ where the third uncertainty is due to the limited knowledge of the branching fractions of J/ψ and ψ(2S) mesons.