Whoa! I have the isolated tracks version of the album, and never heard those organ parts doubling the guitar breaks. And I love being able to hear what George is singing for once!
This is either Bobby Whitlock on Organ and Billy Preston on Piano or Bobby Keys on both Organ and Piano. Judging by the piano's playstyle, my money is on Preston
Whitlock claimed he provided the organ himself. I'm also thinking it was Billy Preston on piano. And yes, George, Eric and Bobby sangs the harmonies together.
Hearing the organ in this track makes it sound like a completely different song! And in a way, it kind of is. The remix makes it sound like an actual song, compared to the massive puddle Phil Spector turned it into.