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It is well known that cool star atmospheres depart from local thermodynamic equilibrium (LTE). Accurate abundance determination requires taking those effects into account, but the necessary non-LTE calculations are often lacking. Our goal is to provide detailed estimates of NLTE effects for FGK type stars for all spectral lines from the ultraviolet to the infrared that are potentially useful as abundance diagnostics. The first paper in this series focusses on the light elements Na, Mg and Al. The code PySME is used to compute curves-of-growth for 2158 MARCS model atmospheres in a wide parameter range. Nine abundance points are used to construct individual line curves-of-growth by calculating the equivalent widths of 35 Na lines, 134 Mg lines, and 34 Al lines. The lines are selected from the ultra-violet to the near infrared wavelength range. We demonstrate the power of the new grids with LTE and NLTE abundance analysis by means of equivalent width measurements of five benchmark stars; the Sun, Arcturus, HD84937, HD140283 and HD122563. For Na, the NLTE abundances are lower than in LTE and show markedly reduced line-to-line scatter in the metal-poor stars. For Mg, we confirm previous reports of a significant 0.25 dex LTE ionization imbalance in metal-poor stars that is only slightly improved in NLTE (0.18 dex). LTE abundances based on Mg II lines agree better with models of Galactic chemical evolution. For Al, NLTE calculations strongly reduce a 0.6 dex ionization imbalance seen in LTE for the metal-poor stars. The abundance corrections presented in this work are in good agreement with previous studies for the subset of lines that overlap, except for strongly saturated lines.