A new generation of coronagraphs to study the solar wind and CMEs are being developed and launched. These coronagraphs will heavily rely on multi-channel observations where visible light (VL) and UV-EUV observations provide new plasma diagnostics. One of these instruments, Metis on board ESA-Solar Orbiter, will simultaneously observe VL and the UV Lyman-$alpha$ line. The number of neutral Hydrogen atoms (a small fraction of coronal protons) is a key parameter for deriving plasma properties such as temperature from the observed Lyman-$alpha$ line intensity. However, these measurements are significantly affected if non-equilibrium ionisation effects occur. The aim of this work is to determine if non-equilibrium ionisation effects are relevant in CMEs and in particular when and in which regions of the CME. We use a magneto-hydrodynamic simulation of a magnetic flux rope ejection to generate a CME. From this we then reconstruct the ionisation state of Hydrogen atoms in the CME by evaluating both the advection of neutral and ionised Hydrogen atoms and the ionisation and recombination rates in the MHD simulation. We find that the equilibrium ionisation assumption holds mostly in the core of the CME. In contrast non-equilibrium ionisation effects are significant at the CME front, where we find about 100 times more neutral Hydrogen atoms than prescribed by ionisation equilibrium conditions, even if this neutral Hydrogen excess might be difficult to identify due to projection effects. This work provides key information for the development of a new generation of diagnostic techniques that aim at combining visible light and Lyman-$alpha$ line emissions. The results show that non-ionisation equilibrium effects need to be considered when we analyse CME fronts. To incorrectly assume equilibrium ionisation in these regions would lead to a systematic underestimate of plasma temperatures.
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