Yes, it's a British term. "A cup of tea" in various local accents becomes shortened into "a cuppa tea". Over decades, this eventually became known as just "a cuppa." So you're right in that there was a next term coming, but it's no longer required, since a "cuppa" is implicitly tea — saying "a cuppa [something else]" would probably cause a mildly confused look followed by a (probably) correct assumption of what you meant. Long live the cuppa. Usually with milk, and sometimes sugar.