Ethan Mordden's recent history of the musical, Anything Goes, ends with an extremely long discographical essay that is essentially a history of cast (and studio-cast) recordings. It starts with The Beggar's Opera and Gilbert and Sullivan and quickly moves to the American musical, charting the first 78 singles of original-cast performers, the first American attempts at albums (with a few original-cast members), Show Boat in 1932 and Blackbirds in 1933. He says The Cradle Will Rock isn't really the first cast album because the show was orchestrated and the album used only piano, as in the theatre. (For some reason he thinks Cradle should only be heard with the original scoring, though the only recording that uses it is very hard to find.)
Mordden also goes into the first sort-of cast albums in the 1930s--for instance, the RCA Victor Porgy and Bess using two white opera singers but the orchestra, chorus, and conductor from the Broadway premiere. Then the emergence of the true cast albums in the 1940s. He says This is the Army was the first and Oklahoma! the second. And so on up to the present day. He also compares discs of, say, Carousel, to show how different they can be in how they present the show. I think he does the same thing with A Little Night Music.