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Westerlund 1 (Wd 1) is one of the most massive young star clusters in the Milky Way. Although relevant for star formation and evolution, its fundamental parameters are not yet very well constrained. We aim to derive an accurate distance and provide constraints on the cluster age. We used the photometric and astrometric information available in the Gaia Early Data Release 3 (Gaia-EDR3) to infer its distance of 4.06$^{+0.36}_{-0.34}$ kpc. Modelling of the eclipsing binary system W36, reported in Paper II, led to the distance of 4.03 ± 0.25 kpc, in agreement with the Gaia-EDR3 distance and, therefore, validating the parallax zero-point correction approach appropriate for red objects. The weighted average distance based on these two methods results in dwd1  =  4.05 ± 0.20 kpc (m − M  =  13.04$^{+0.11}_{-0.12}$ mag), which has an unprecedented accuracy of 5 per cent. Using the Binary Population and Spectral Synthesis (BPASS) models for the Red Supergiants with solar abundance, we derived an age of 10.7 ± 1 Myr, in excellent agreement with recent work by Beasor & Davies (10.4$^{+1.3}_{-1.2}$ Myr) based on MIST evolutionary models. In Paper II, W36B was reported to be younger than 7.1 Myr, supporting recent claims of a temporal spread of several Myrs for the star-forming process within Wd 1 instead of a single monolithic starburst episode scenario.