As a Nottm city centre born & bred person, I'd call it Suth'ell, but I know a lot (most ?) locals call it Southwell. I don't know if this is "the native way" to say it, or if it started when the village became well to do (or, being a major diocese, maybe there was always this split). Must admit, if I was talking to a non local now I'd probably say Southwell, because I wouldn't want to be taken as an uneducated slob....instead I could be taken as an unmitigated snob.
But I really have no idea which is "correct"