The version of Allegri's Miserere that we are accustomed to hearing was the creation of a 19th century musicologist according to Graham O'Reilly in a book to be published this year (See: Allegri’s real Miserere revealed!)
Graham O'Reilly is the director of the Ensemble William Byrd which is responsible for the recording in the above YouTube video. The Product Description on Amazon says:
Those for whom the Allegri Miserere stands as a choral music icon will find this recording something of a shock. Getting on for a century after Gregorio Allegri penned his masterpiece in Rome in 1638, his work was emulated by Tommaso Bai, maestro of the St Peter's Basilica choir. To cut the story short, Bai's and Allegri's Misereres were eventually melded into a composite work for performance in the Sistine Chapel. A copy of this extraordinary hybrid was preserved in the Vatican archive after the papal choir was dissolved in 1870. From this, Hugh Keyte produced the performing edition heard here, incorporating the dramatic style and expression known to have been employed--including seemingly bizarre portamenti which at first seem like a joke.The Description goes on to suggest that some may feel queasiness at this performance while others will be fascinated. I confess to being fascinated, indeed enchanted.
Archiv Music has a longer review by David Vernier. Thanks to Fr James Bradley on Facebook for posting the video. Fr Bradley blogs at Thine Own Service.