I would guess that the speaker is from the Southern U.S., where the pace of life is slower than the rest of the country. "All the time in the world" by itself gives a sense of unlimited time for whatever is happening, but adding "southern" gives it a different flavor. It would suggest an even greater sense of leisure, but also calls attention to the speaker's culture (perhaps not shared with "my dear"). There might be various subtle implications depending on the context. I'm imagining wry humor in your example.

I live in the south of the USA, so I think probably that means "plenty of time" because southern people are known for being pretty laid back.

Without further context, it doesn't make sense. It sounds like the continuation of some dialogue: Her: "Do we have time?" Him: "All the southern time in the world, my dear." It still doesn't make sense but he's saying that they do not have a time limit. They have a lot of time, as much time as they need or want.