'my daughter' sounds incredibly formal and old-fashioned. Other than that I think most people tend to just use names.
For a standard 'sweet' name, a family might use something like 'dear', or 'darling', 'honey' or 'love' (usually without the 'my'). I couldn't say which would be most common and some families may never say any of these. Brits are not a very soppy lot in general.
For a nickname, there are countless options... and being British quite often they are a bit cheeky/rude or based upon some silly family event from years ago. 'Nona' is actually my family nickname coming from one of my niece's mispronounciation of my real name when she was a baby. One of my friends is called 'Lurch' or 'Big Bird' by many, including most of her family, as she is very tall and they were names of tall fictional TV characters!