I don't know a lot about musicals, but it felt like each song was referencing a musical sub genera. Nurse Chapel's song seem to reference something jazzy like Chicago, another song sound like it would of fit into the 60s Marry Poppins/Sound of Music.
Most of the songs are in a modern Broadway poppish sounding Pasek & Paul vein (La La Land, The Greatest Showman, Dear Evan Hansen) with the exception of Una and Kirk’s “Connect to Your Crew”, which is more old school. Maybe they were going for a Gilbert & Sullivan vibe, but it’s closer to Rodgers & Hammerstein in The King and I and The Sound of Music, musically.