While the 30-06 is certainly a good choice for all the above-mentioned game (including nilgai), it is not the only suitable caliber for the job. If you are planning on hunting all of this game with one rifle loaded with one particular round, then I would go with something .264 or better bullet diameter, a performance bullet that will hold together but expand well, and any caliber that can push at least a 140-grain bullet at about 2500 fps or better. Most important, however, would be a rifle that you are confidently accurate with out to 200 yards, with a scope for precise shot placement.
I know that covers a lot of ground, but the type of exotics you will likely encounter are all elk-sized or smaller, and the turkeys will require absolute precision accuracy. So as long as you are on the higher end of the whitetail calibers, you should be fine. Good luck on the hunt! :grin: