Red osier stays bright white for a long time. At least mine did. But I've heard that Eastern Woodlands Indians preferred arrowwood viburnum for their arrows. This wood is not as bright as red osier and sometimes becomes gray-ish. It's also heavier and harder than red osier.
The color of the arrows is also affected by what you use for a finish coat. Animal fat can be mixed with finely ground pigment of any color and applied. But I've recently read an account (in a museum in Vermont, I think) that the "oil" from the castor gland of a beaver was commonly used as water repellent. I don't know how this would affect the color of the wood, though.