BA really should just put Business Class on all documentation, they can add CE/CW if they like, but I bet a good proportion of passengers don't even know what Club Europe/World means.
I don't think the word "Club" is particularly indicative of a cabin, even if your first language is English... and what does being in the EU have to do with whether a generic airport staff member should know that BA chooses to use special names for its J class?