Bloom won the Academy Award for Best Film Editing and the American Cinema Editors (ACE) Award for Best Edited Feature Film for Gandhi (1982). He was nominated for Academy Awards in 1981 for his work on The French Lieutenant's Woman and in 1985 for A Chorus Line. He was nominated for the BAFTA Award for Best Editing in 1981 for The French Lieutenant's Woman, in 1982 for Gandhi, and in 1984 for Under Fire (with Mark Conte).
In 1999, the ACE awarded Bloom its Career Achievement Award.
In 2001, he won the Primetime Emmy Award for Outstanding Single-Camera Picture Editing for a Limited Series or Movie for Wit, and, in 2004, the ACE Award for Best Edited Miniseries or Movie for Non-Commercial Television for his work on Angels in America (2003).