First, the basic context of this paragraph is in a chapter on the gradual introduction of French-derived words into Middle English by a Norman (French) aristocracy. So the royal family would be the French-speaking rulers of England from 1066 until they switched to learning English as a first language in the 15th century.
The basic structure of the sentence is interrupted by a long parenthetical statement, a phrase or clause that glosses "the royal family." Without that long parenthetical, we would have:
Even that great redoubt of French, the royal family, began to take notice.
"That great redoubt of French" is the noun phrase serving as subject, "began to take notice" is the predicate, and "the royal family" is another parenthetical statement (an appositive) grammatically describing "that great redoubt of French."
The long segment describes a pertinent quality of the royal family. Grammatically, a parenthetical doesn't change the syntax of what's around it. However, they can be complex, and Bragg has used this one to describe how he thinks the royal family sits in relation to the English language of its subjects:
the royal family, unbelievably slow in appreciating their good fortune
So the royal family was slow to appreciate their good fortune. What was the source of this good fortune? The answer comes in two parts that are logically connected. The second part is trickier, so I spend more time explaining it:
in ruling the country they did with the language it was relentlessly replenishing,
They (the royal family) ruled the country they did (England). Not every country would blend its own language with French to create "astonishing precision and flexibility."
With the language it was relentlessly replenishing. First, it is slightly unclear what "with" goes with. Second, the most sensible correspondent to "it" is near the start of the sentence.
"With the language it was relentlessly replenishing" can modify ruling, so that the royal family ruled England with the language (French) it (England? English?) was relentlessly replenishing. I reject this reading because the action taken by the probable "it" wouldn't make sense: English didn't replenish French, but rather French vocabulary replenished English.
"With the language is was relentlessly replenishing" modifies "the country (they did rule," such that the country (England) has the language (English) that it (French, which the royal family is the great redoubt of) was relentlessly replenishing.
The second reading may be a stretch for reading "it," but note examples in the paragraphs above where "it" is left rather fuzzy: "that's the beauty of it" at the start of a previous paragraph has a general sense of "English" or the English embrace of French loan-words, or something like that. Since the author seems otherwise inclined to use "it" in suggestive ways dependent on context rather than syntactic placement, "French" can be the referent for "it."
In short, the (Norman-descended, French-speaking) royal family in England was slow to appreciate that it ruled a country whose people would expand the expressive capacity of English by adapting the ruling family's French words into English while keeping the basic structure and vocabulary of English. Nonetheless, the royal family eventually noticed. Bragg may have in mind how the royal family became patrons of English authors (like Geoffrey Chaucer and John Lydgate) and switched to speaking English as a first language.