Gregory Martin was chiefly responsible for the translation. The entire Bible was translated before his death in 1582. Others at the Douay College (Then removed to Reims because of warfare around Douay) proofed and annotated his translation. It has been noted by experts that Martin used an unknown Vulgate of remarkable exactness and accuracy. Consider that the Clementine edition of the Vulgate was still in process then. Martin was a Hebrew and Greek scholar, so to answer your question directly, it was in the period of 1560-1580 that the D/R was being “diligently compared.”
Wikipedia (and others) tend to accuse the D/R of excessive Latinisms and difficult English. And to be sure, by the time Dr. Challoner revised it in the mid-18th century, that was certainly true. But in its own time, the Latinisms were heavily annotated to explain their use, and MANY of them wound up entering into the English vocabulary because of it. It is important to always keep in mind that the OVERRIDING concern of the Rheimists was that the Protestant Bibles (particularly the Geneva and Bishops bibles) were purposely mistranslated to support their doctrines. Therefore the Chemists wanted to counter those translations with one that was very literal and true to the most accurate Vulgate. It gave the English Catholics some foundation for understanding the subtle and egregious mistranslations then floating around everywhere in England. IOW, it accomplished it’s primary mission perfectly, even if it wasn’t as smooth a translation as the later King James Version (which borrowed much from the D/R.)