The Ladywell was near or in Wrottesley Street, and Ladywell Walk is still there now. I think it was only a public source of water, I don't think it was a bath, although I would not swear to that. No idea when it ceased or if it has been culverted. I have got information from a book which says 'Water from the well or spring helped the fill the moat around the parsonage'.
The book is of street names and it has no mention of Little Cannon Street, only Cannon Street and Cannon Passage. I am sure someone else will have information. It certainly does not seem to be marked on the map of 1902.