'Diglossia' has a significantly different meaning in English.
"a situation in which two languages (or two varieties of the same language) are used under different conditions within a community, often by the same speakers. The term is usually applied to languages with distinct “high” and “low” (colloquial) varieties, such as Arabic."
Διγλωσσία means the same thing in Greek, too. Being δίγλωσσος is another thing, and means "bilingual".