As usual with organic materials, it depends. Some osage can be banana yellow at first. Within a few days unfinished, it'll be much less bright. But, a finished bow, with a good plastic finish will stay yellow for years if it is kept out of the sunlight. If you want the darker color sooner, leave it unfinished and in the sunlight for a few weeks. If it is already finished, it'll take longer but leave it in the sun and it'll get a tan pretty quick anyway.
The darkening is an oxidation reaction so a wash down with sodium hydroxide (lye) before finishing will get the color right away. I like to watch them get darker over time. Really old osage can be almost black, or purplish.