Iron plays a crucial role in studies of late-type stars. In their atmospheres, neutral iron is the minority species and lines of FeI are subject to the departures from local thermodynamic equilibrium (LTE). In contrast, one believes that LTE is a realistic approximation for FeII lines. The main source of the uncertainties in the non-LTE (NLTE) calculations for cool atmospheres is a treatment of inelastic collisions with hydrogen atoms. We investigate the effect of FeI+HI and FeII+HI collisions and their different treatment on the FeI/FeII ionisation equilibrium and iron abundance determinations for Galactic halo benchmark stars HD 84937, HD 122563, and HD 140283 and a sample of 38 very metal-poor (VMP) giants in the dwarf galaxies with well known distances. We perform the NLTE calculations for FeI-FeII with applying quantum-mechanical rate coefficients for collisions with HI from recent papers of Barklem (2018, Cat. J/A+A/612/A90B18), Yakovleva, Belyaev, and Kraemer (2018CP....515..369Y, YBK18), and Yakovleva, Belyaev, and Kraemer (2019MNRAS.483.5105Y , YBK19). We find that collisions with HI serve as efficient thermalisation processes for FeII, such that the NLTE abundance corrections for FeII lines do not exceed 0.02dex, in absolute value, for [Fe/H]~>-3 and reach +0.06dex at [Fe/H]~-4. For a given star, different treatments of FeI+HI collisions by B18 and YBK18 lead to similar average NLTE abundances from the FeI lines, although there exist discrepancies in the NLTE abundance corrections for individual lines. With using quantum-mechanical collisional data and the Gaia based surface gravity, we obtain consistent abundances from the two ionisation stages, FeI and FeII, for a red giant HD 122563. For a turn-off star HD 84937 and a subgiant HD 140283, we analyse the iron lines in the visible and the ultra-violet (UV, 1968 to 2990{AA}) range. For either FeI or FeII, abundances from the visible and UV lines are found to be consistent in each star. The NLTE abundances from the two ionisation stages agree within 0.10dex, when using the YBK18 data, and 0.13dex in case of B18. The FeI/FeII ionisation equilibrium is achieved for each star of our stellar sample in the dwarf galaxies, with the exception of stars at [Fe/H]~<-3.7.