What word describes the day between two statutory holidays—"not officially active"?

Your question is unclear. Let me see if I understand it correctly. To say 'not officially active' suggests that the school is unofficially active (such as a teacher tutoring students at the school even though it's a Saturday).

In your example, you stated that Tuesday is an official holiday and Thursday is an official holiday. Therefore, no one will attend classes at the school on Tuesday or Thursday.

English speakers would say:

"There is no school on Tuesday."

"There aren't any classes on Tuesday."

or, more formally,

"Classes are not in session on Tuesday."

EDIT:

I don't think there is a idiom like that in English. English speakers would create an excuse on the spot, like"

"Oh, there is no class today because the teacher is sick."

"There is a school assembly, but I didn't have to go."

Any plausible excuse will do.

In the case of school, one might call Wednesday an unofficial skip day. When taking the day off work, you might call it a floating holiday. In general, though, in the United States the only days that come close to this situation are the day before and the day after Thanksgiving, and I have not heard any word or phrase used consistently to describe them in the way you are asking.

In the US I've heard my sixteen-year-old son explain it this way:

I don't have to go on Friday because it's the last day of classes before the vacation and no one's going to really do anything that day.

I think your example is probably going to result in no one attending classes for the whole week. A better example would be an official holiday occurring on a Tuesday. Then people would often take Monday off too. In this situation, people would say

Everybody's going to be taking Monday off for a long weekend.

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.