I'm a big Desplat fan but for some reason hadn't heard of this (probably because I'm not a Malick fan at all), so I'm glad for this thread bump. I'll give it a listen.
I just watched the extended Criterion 3 hour cut of the film, and it reminded me of something I've been wondering since the film first came out and I don't think has been clearly answered.
Did ANY of Desplat's score end up in the finished film as it was released in 2011?
Watching it again, especially after watching the music featurette on the Criterion disc, I kept realizing that every piece of music seemed to be a (beautifully chosen) existing piece.
Apparently a few cues in the finished film are by Hanan Townshend, who would go on to write music for Malick's next four films.
I did notice two cues from Desplat's score over new material in the extended Criterion cut - "River" over a sequence 20 mins into the film showing a mix of archival B&W footage, then Sean Penn wandering through various spaces and briefly getting in a series of fights, and "Childhood" over a sequence where Pitt's character narrates moments from the life of his own father, about 1hr18mins in. I confirmed the cues by Shazaam on my phone.