There were Nazi sympathizers in what would have been Union states during that time. Support for the war was not 100%.
If we're defining the South by the slave trade, then they'd have strong ties to the British, the French, and the Dutch through that. All Allied powers. Connections to the Axis powers would have been Portugal and Spain.
You're making the mistake of assuming that just because one immoral act was committed that there were no redeeming values to be found in the people.
That is just untrue.
It is interesting to speculate what would have happened had secession succeeded. We can speculate that slavery would have still eventually ended, maybe later. Reconstruction obviously doesn't happen. Do the two nations become more Canada/US, Mexico/US, East/West Germany, or even North/South Korea? Are there scuffles over unclaimed territory? Who knows.
The point is that it could have just as easily be the former Union joining the Axis. We don't know.