Based on the European rules, they might have some success claiming exclusivity on ‘Atlanta cola’ (though Coca Cola probably wouldn’t want them to do this, as any company making it in Atlanta to a specified standard would be able to use the name).

Honestly I assume this is more or less why the food industry resents protected names so much; it’s perceived exclusivity/specialness, but not based on a brand controlled by a single company.

Their main purveyor of cola will probably disagree, since they want to make cola everywhere

First they're already strongly protected by trademark law, second they make their Cola pretty much everywhere (except maybe the base syrup), third there's already a bunch of Cola (ever heard of Pepsi Cola?).

Atlanta doesn't have a multiple hundred years tradition of making "Cola" that started before the industrial age, which is the case for most EU food concerned by this.