The perfect conjunctive (= subjunctive) of reverti would be reversi sint, not reversi fuerint.
The perfect forms of deponent verbs are typically formed with present and imperfect forms of esse, not perfect and pluperfect forms from the stem fu-.
Therefore you have a "double perfect", much like saying "they have had returned" in English.
While rare, the double perfect appears in classical Latin.
It emphasizes the completed nature of the action: the people have first completed returning and then started to approach.
I have a feeling that this construction became more common after the classical era, but I have nothing to back this specific claim up with.
Note that the text is not classical, so classical grammar is not the perfect reference.
The main question seems to be about mood: why conjunctive (= subjunctive) instead of indicative?
Following the sequence of tenses in a subordinate clause, the perfect tense is the only option for conjunctive.
And perfect is indeed the most suitable choice for indicative as well.
In general, conjunctive is found very often in temporal clauses.
I'm not sure if I'm reading Allen and Greenough's temporal clauses right, but they seem to imply that temporal clauses with quando (almost) always take conjunctive instead of indicative.
They classify this is a conditional relative clauses, and that makes sense: the subordinate clause expresses a condition.
When the statement of the subordinate clause is not a logical fact but rather a thought — typically of the subject of the dominating clause — then a conjunctive is always used.
See A&G on intermediate clauses.
That is, the singing takes place when the singers think they have returned.
If what matters is the thought, the conjunctive is the choice, whether or not the event is factual.
Do bear in mind that most texts — and anything I say — on subordinate clauses concerns the classical form of the language.
The text you quote is more recent, and practices shift over time.
Varieties of Latin
The double perfect is valid but rare in classical Latin.
I have understood that it is much more common in the Vulgate, which in turn has influenced medieval Latin to a great extent.
I don't know where Vulgate got its influence; probably Greek and Hebrew.
See this question and its answer for more examples of Vulgate effecting subordinate clauses in medieval Latin.
The double perfect is unusual but grammatical even in classical Latin, albeit rare.
The conjunctive mood is expected.
The text is medieval rather than classical, and I don't know whether these choices are more common in that era.