Advances in merged-beams instruments have allowed experimental studies of the mutual neutralisation (MN) processes in collisions of both Li$^+$ and Na$^+$ ions with D$^-$ at energies below 1 eV. These experimental results place constraints on theoretical predictions of MN processes of Li$^+$ and Na$^+$ with H$^-$, important for non-LTE modelling of Li and Na spectra in late-type stars. We compare experimental results with calculations for methods typically used to calculate MN processes, namely the full quantum (FQ) approach, and asymptotic model approaches based on the linear combination of atomic orbitals (LCAO) and semi-empirical (SE) methods for deriving couplings. It is found that FQ calculations compare best overall with the experiments, followed by the LCAO, and the SE approaches. The experimental results together with the theoretical calculations, allow us to investigate the effects on modelled spectra and derived abundances and their uncertainties arising from uncertainties in the MN rates. Numerical experiments in a large grid of 1D model atmospheres, and a smaller set of 3D models, indicate that neglect of MN can lead to abundance errors of up to 0.1 dex (26%) for Li at low metallicity, and 0.2 dex (58%) for Na at high metallicity, while the uncertainties in the relevant MN rates as constrained by experiments correspond to uncertainties in abundances of much less than 0.01~dex (2%). This agreement for simple atoms gives confidence in the FQ, LCAO and SE model approaches to be able to predict MN with the accuracy required for non-LTE modelling in stellar atmospheres.
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