According to Frederick Allen's history of Coca Cola, "Secret Formula" (fun read), the African Kola Nut was the source of the caffeine incorporated along with the original cocaine, "Coca" as in the coca leaves cocaine is refined from, to make it an effective headache remedy just as Canadian aspirin has caffeine in it too for superior headache relief. It's got about the same amount of caffeine as a single cup of coffee and coffee was already a major beverage in U.S. culture, on par with tea by then. The coffee craze and discussions about regulating it are 17th Century England, Holland, etc.-coffeehouses like Lloyds of London.
The Kola Nut's main contribution was being sooooooo bitter that it took an enormous amount of cane sugar blended in to make Coca Cola headache remedy drinkable and after enough sugar and carbonation with traces of lemon and lime juice, cinnamon, cloves, etc. actually palatable and tasty, just like many other patent medicines and cough syrups did at the time. Overcoming the Kola Nut to keep some caffeine in there made Coca Cola the most sugary beverage available which is probably how it became such a popular recreational beverage instead of staying a competitor to aspirin.
Pepsi Cola's main claim to fame was doubling the amount of sugar by volume over Coke. The Kola Nut wasn't available thanks to U-Boat disruptions in African shipping so the caffeine source changed in World War II but not the high cane sugar content.
By World War I it's a sugary beverage for pleasure, not a medication. So probably no impact at all.