Sentence Correction?

You can rephrase the sentence as:
By the mid-1700s, the French government had fully realized that the success of its state building projects depended on the welfare of its people, and it began a concerted effort to prevent famine and plague.
The answer is A.
Answer B could not be corrected, as "until" has no place in this context.
Answer C should be corrected as:
In the mid-1700s the French government began a concerted effort to prevent famine and plague, when it had fully realized ...
Answer D should be corrected as:
The French government began a concerted effort to prevent famine and plague in the mid-1700s, having fully realized ...
Answer E should be corrected as:
When the French government began a concerted effort to prevent famine and plague in the mid-1700s, it had fully realized ...

1.) timeline action of past perfect tense
There are several usages of past perfect tense and the specific use in this sentence is about :
- Duration Before Something in the Past for Non-Continuous Verbs (i.e. the stative verb realize)
O(had realised) --------(duration)--------X(began) the second point of action in the past which happened in mid-1700s

typical example from a grammar book:
By the time Alex finished his studies, he had been in London for over eight years.
"By" refers to [possibly before and no later than X point of time]
(note: duration of state can be implicit)

Now applying the structure to this sentence:
By "the time" (that) the French government began a concerted effort to prevent famine and plague, it had realized (for some years) that ....
As the point of time is about the mid-1700s, we can further construct the sentence as:
By the mid-1700s, "when" the French government began ..., it had "fully" realized that .....

2.) By ("no later than") vs Until ("up to")
until = up to the point in time (something continues to happen but stops at the point of time)
not until = not before a particular time or event

Rewriting the sentence with "not until":
It was not until the mid-1700s that the French government (finally) began a concerted effort to prevent famine and plague, after it had "fully" realized that the success ...
(It may refer to a plan or willingness, rather than just a time sequence of two events as expressed in the answer A).

*** rewriting with until
The French government did not begin a concerted effort to prevent famine and plague, until it had fully realized that the success ..
(the timing is different from the answer A)
.
3.) perfect participle construction as indicated in Kevn's answer D
Having fully realized that ..., the French government began a concerted effort to prevent famine and plague in the mid-1700s.

Remember, participle phrases give information about condition, reason, result or time.
Yes, we can use perfect participle to emphasize that one event happened before another. However, it could also imply a cause-and-effect relationship.

1. time (as in a complete adverbial clause)
After it had fully realized that ..., the French government began a concerted effort to prevent famine and plague in the mid-1700s.

2. result (as in a complete adverbial clause)
Because it had fully realized that ..., the French government began a concerted effort to prevent famine and plague in the mid-1700s.

By the mid-17th century the French government began a concerted effort to prevent famine and plague, when it fully realized that the success of its state building projects depended on the welfare of its people. Yip