Elfman composed the theme but his orchestrator/conductor from Batman, Shirley Walker composed the music for the show. I say this because Walker is an extremely underrated composer and rarely gets her due. In addition to The Flash, she scored the entire WB DC animated shows including: Batman TAS, Superman TAS, Batman Beyond, and Justice League. Her theme for Superman to this day is still my favorite Superman theme.
Unfortunately, she passed away in 2000 well before her time.