The first real "Cast Recording" was done three years before "Anything Goes" with the Astaires, during the run of "The Band Wagon." It's only 20 minutes long but it covers most of the show (including the overture) but not with the show orchestra.
Traditionally, Victor would record a few solo numbers from the hit shows but usually with studio bands of a famous bandleader. In England, they often used the pit orchestras so the show music that IS available from London back then is more authentic.
http://www.broadwayworld.com/board/readmessage.cfm?thread=972787#3631451
http://www.broadwayworld.com/board/readmessage.cfm?thread=963561#3533883
http://www.broadwayworld.com/board/readmessage.cfm?thread=955158#3440952
http://www.broadwayworld.com/board/readmessage.cfm?thread=954269#3427915
http://www.broadwayworld.com/board/readmessage.cfm?thread=955012#3441622
http://www.broadwayworld.com/board/readmessage.cfm?thread=954344#3428699