If you're just looking for a natural protectant for something that will live a life of ease mostly indoors, I'll second the Feed-N-Wax.
If you're looking for a UV protectant and sealer that brings the life out of wood grain, I don't know how much you want to spend (though you probably only need a 250ml can), but for a varnish, Epifanes is at the top. I used it on the deck of a wood kayak after reading about how difficult it is to varnish and it could not have been easier or had better results (minus a single drip under the coaming) just following the directions on the can. It's a marine product, so it obviously can withstand whatever this bike would go through and it uses the best ingredients. The kayak lives in the garage, but it's spent months outside, has been in the ocean, and banged around, and I think it's going on 8 years. The paint is beat up, but the varnish still looks good. On the other wood kayak deck, I have automotive clear coat and the difference in appearance is dramatic, with the CC looking lifeless next to the glow of the varnish. For stain, Varathane or Zar go on incredibly smooth and easy and blend without effort or care.