  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.
