TL;DW:

The flavor mostly comes from essential plant and spice oils such as nutmeg, cinnamon, lemon, lime, and quite a few others.

Surprisingly, there seems to be a small amount of vinegar that's essential to getting the correct flavor.

There's also a mandatory heating step that allows the phosphoric acid to "develop" some of the raw flavor compounds from the essential oils into more complex compounds with different flavors.

In the end, it's all stuff you can buy yourself. The start-up cost of getting all the ingredients and equipment is high, but once you have them you'd basically be set for life and could make your own coke for much cheaper than buying the "real" stuff at the store.