"Normal Coke" is usually used when trying to differentiate between "Diet Coke" and the original version.
All the wording used in fast food restaurants is generally American-based as that's where the fast food places came from. The same goes for "fries". These kind of fried potatoes are, of course, called "chips" in the UK, but when ordering them at these kinds of places, we always call them "fries".
I would never expect to hear someone order "normal Coke and fries" in either country, only "regular Coke and fries".