There was a discussion on TMP some time ago where one of the commenters advanced the idea that for more durable items such as spear shafts, ax and mace handles, etc. that with age they would become darker if they were oiled and polished or had a preservative applied. Conversely javelins and arrows and other missile weapons would be a lighter color as they would be expended before aging to any degree.
I originally used the Model Masters 'Wood' for any items made of wood but after that discussion switched to using a couple of darker shades of brown - espresso and burnt sienna in craft paints with a more medium brown thrown in for variety for the lances, etc. and lighter shades such as khaki for javelins, arrows, etc.
After reading that yew and elm used in bows was heartwood they get the darker shades as well.
I have been using khaki as well for the bamboo bows of my ancient Indians.
For wood that isn't being treated I use lighter shades based on years of having tools with wooden handles exposed to the elements.
It really depends. I've tried to use a variety of wood colours inside the same army and the same element, trying if I can to get info on what sort of wood a people would have used historically. One of the most recognisable examples is my Tlingit army, the bows and spears of which I painted with a particular orange tinge to represent wood from Thuja plicata, the "red cedar" (actually a cypress) from Alaska:
My 'standard color' for newly-cut wood - javelins, etc - is Delta Ceramcoat 'Spice Tan', #02063. For 'seasoned' wood it varies. Have a 'fortified camp' sitting on workbench, every time I use a shade of 'brown' paint, daub a yew of the 'stakes' (scribed plastic) that color.
Meanwhile the days lengthen and warmer grow, and atavistic drives whisper 'time to clear workbench, spend days just cleaning the flash from new figures, sanding the bottoms, washing them, for soon 'twill be warm enough to go outdoors and Spray Primer Around!'
I was using the Derivan Minis colour "Spear Brown" which seemed made for the purpose until I ran the bottle out. Derivan Minis are no longer produced a victim of their own success - I bought most of my large collection of Derivan Minis between 2006 and 2009 and most are still useful - the 55ml bottles do not dry out.
I moved to a burnt sienna that was close but lo and behold when I opened up a plastic container of Spare Paint (to put some tubes that the younger of the (not so) little warbands had used in a school project) I found a pristine bottle of Derivan Spear Brown ready and waiting.
Because it has my troops all over it. THAT makes it MINE!!!!
I typically go for a lighter brown just so spears and other important woody things stand out a bit at arms length and don't fade into the background. I also try to do the same thing with the actual figure and the base, contrast them enough so the figure pops out of rather than fades into the base.