Lincoln green originated in Lincoln, England around 1510 and was made with blue and yellow, most likely plant-based, dye. I'd say woad for the blue and weld for the yellow. Lincoln green could have actually been considered a little expensive though so for him to have it I would say his would have to be flawed, less refined or stolen. Which is pretty on track for him lol.
Assuming that the color Will is wearing is actually scarlet, which it most likely is... It's definitely old enough to be considered historically accurate. Though whether or not he could afford it, like with Robin Hood, is the question. Scarlet dye is from around 3000 BCE and it was typically made with bugs. One type of bug was the kermes which was a beetle is the oldest method as far as I know. Scarlet dye was expensive so in order for it to be historically accurate for him he would have had to steal it, since less refined versions would have also probably been out of his price range, which is probably not out of character.
So yes, I would say that both are historically accurate and not modern movie magic ✨