According to the Collins Dictionary, the noun "graduate" slightly differs in meaning with regard to the location of speaking:
In Britain, a graduate is a person who has successfully completed a degree at a university or college and has received a certificate that shows this. – "They are looking for graduates with humanities or business degrees."
In the United States, a graduate is a student who has successfully completed a course at a high school, college, or university. – "The top one-third of all high school graduates are entitled to an education at the California State University."
The difference is seemlingly provable in the case of the verb "to graduate" as well:
In Britain, when a student graduates from university, they have successfully completed a degree course. – "She graduated in English and Drama from Manchester University."
In the United States, when a student graduates, they complete their studies successfully and leave their school or university. You can also say that a school or university graduates a student or students. – "When the boys graduated from high school, Ann moved to a small town in Vermont"; "In 1986, American universities graduated a record number of students with degrees in computer science."
From the information above I would come to the conclusion that in British English the discussed noun and verb cannot apply to secondary education, only higher education. Is that correct? And if so, what similar noun(s) and verb(s) can be used in BE to refer to finishing secondary education?