What finish is on the current wood? You may be looking at less work and better results if you buy new unfinished wood, finish appropriately, and replace existing stuff. Especially if the wood has been regularly treated with something like Pledge or other polishes....
Check the finishing/refinishing section on this sticky -
The Stock Projects Mega-Sticky! - RimfireCentral.com Forums . Yes, it is for spoons but George (noremf) is a fount of knowledge when it comes to wood, finishes on wood, how to remove them, etc.
More from George/noremf on the bad stuff in various polishes, etc -
The School of Woodworking Knowledge (questions? try here!) - RimfireCentral.com Forums