I live in Chile, not in England, but English is my mother tongue. However, the words that I know for the situation that you describe are used in Spanish. For a good example (from Wikipedia):
In Chile, a "sandwich" is a day that falls between two holidays, independently of whether it's a holiday by itself or not. In the latter case, workers may take it off on account on vacation days, an action called "tomarse el sandwich" (lit.: "taking the sandwich"). In formal writings, the term "interferiado" is used instead of "sandwich". In colloquial contexts, these days, almost always a Monday or a Friday, may be called "San Lunes" or "San Viernes" (lit.: "Saint Monday" and "Saint Friday", respectively) as well.
We might say: "Tuesday is Independence day; I am going to make Monday a sandwich and install the kitchen cabinets."
It is called a "sandwich" day because it is something of a different sort between two other things, like meat between two pieces of bread. "Sandwich" is native English, but I have no indication that it Is ever used that way in England.
The English are rather more cool and disciplined in their approach to holidays, as indeed they traditionally are in their approach to many other aspects of life. Indeed, the article sited above also states:
This is typically referred to by a phrase involving "bridge" in many languages; for example in some Spanish-speaking countries the term is puente ("bridge") or simply "fin de semana largo".
Four-day bridge weekends are commonplace in non-English speaking countries, but there are only a couple of examples in English-speaking countries:
Long weekend, sandwich or bridge, it is usually something that the student or employee "takes" or "makes" on account of leave days, rather than the institution.