It was some years since I asked at the old Central Library and I was actually enquiring about the records of the Asylum for the Infant Poor, which was closed in 1852. The answer I got was that nothing survived for that institution, but they had the workhouses from 1880. As I had no interest in the workhouses, I didn't press for further details. Of course, what they have and what you can actually get your hands on, because a lot of records are in such poor condition, may be two entirely different things!
Maurice