A complete theoretical analysis of the $C$-conserving semileptonic decays $\eta^{(\prime)}\to\pi^0l^+l^-$ and $\eta^\prime\to\eta l^+l^-$ ($l=e$ or $\mu$) is carried out within the framework of the Vector Meson Dominance (VMD) model. An existing phenomenological model is used to parametrise the VMD coupling constants and the associated numerical values are obtained from an optimisation fit to $V\to P\gamma$ and $P\to V\gamma$ radiative decays ($V=\rho^0$, $\omega$, $\phi$ and $P=\pi^0$, $\eta$, $\eta^{\prime}$). Theoretical predictions for the six decay widths and associated dilepton energy spectra are calculated and presented.
In addition, we investigate the prospect of observing new-physics signatures via $CP$ violation in $\eta^{(\prime)}\to\pi^0\mu^+\mu^-$ and $\eta^\prime\to\eta\mu^+\mu^-$ decays at the REDTOP experiment. We make use of the SMEFT to parametrise the new-physics $CP$-violating effects and find that the projected REDTOP statistics are not competitive with respect to nEDM experiments for these particular processes.