Wam
It is a bit of an exaggeration to say that prostitution had been available for a century. When the houses were built they were respectable residences. A few to the west of the Rae were I believe part of Edgebaston, and these were definitely middle class. If you look at the censuses around the turn of the century you see many were occupied by owners of businesses , respectable widows and professors of music (which is not the same as having a "music teacher" card in the local shop), and similar. A little further east were a number of significant businesses, and further east more working class , but still respectable housing. I think the change occurred gradually, as Phil says, probably around world war 2. After that the area did became notorious, especially Varna Road, which was a little embarrassing when a friend (with a rather plummy accent) in around 1966 came for an interview with Midland Red, and his request for Vernon road was interpreted by the taxi driver (with strange looks) as Varna Road.
The same has happened to a number of similar areas in other cities. Leeds had/has Chapeltown, which was a very prosperous jewish businessman's area originally, while St Paul's , in Bristol, was similar