I have a death certificate that was (I think) 1 back Raddlebarn Road, which was Kings Norton Workhouse. That was in 1924.
I believe that in the early 20th century, a law was passed that forced registrar's to put down the address of the workhouse as a number and road, basically to stop Workhouse being on the certificate's. People who needed medical help, but could not afford it, often chose to enter the workhouse, so that they could use the worhouse hospital. I have looked through the records for Burton Union Workhouse in Staffordshire, and there is plenty of cases were people enter the workhouse, are sent to the hospital, and either die or leave soon afterwards.
In 1930 (or there abouts), Neville Chamberlain (a local man!) passed a law that rewrote the poor law system. The workhouse were closed - children went to orphanges. The problem was, they could not get rid of the adults (and possibly some children) already in the workhouse, and I have no idea how long it would have taken for them to stop taking new cases in.
When Clement Attlee won the 1945 general election, he set up the NHS, which took over many of the old workhouses to use as hospitals. It is very likley that this is when Selly Oak Hospital was born, although I can not prove/disprove that.
Tippin
PS: Kings Norton was in Worcestershire. Worcestershire used to bend round into Warwickshire, so that places on the east of Birmingham, like Yardley and Moseley, were in Worcestershire. In fact, Birmingham used to be in three counties - Warwickshire, Worcestershire (to the south, and parts of the east), and Staffordshire (to the west).