I doubt that you'll see a wooden bike at a professional level anytime soon. Wood has a low strength to weight ratio compared to metals or carbon fiber. It's also highly variable in it's composition. You have to use more of it to insure that you overcome that variability.

It also has some inherent problems with strength in different directions. In compression, i.e. pushed on from top to bottom, wood is very strong. That's why it's great stuff for studs in walls where it's used with the grain of the wood. But pushed on from the side. like you would for a down tube, it's not as strong by a huge factor. That's why studs are used standing up and not laying down. If you are going to use wood as a header in a wall, for example, you have to support it every 16" or make it really, really thick. A floor beam made of straight grain wood, for example, needs to be 12" by 18" to support a floor load. Even then, you have to orient the beam in a specific direction to gain the most strength.

You could reinforce the wood with various compounds but you still can't thin it out like you can with carbon fiber and retain strength.