The Afrikaaner Pure Nationalist Party which was the chief oppostion to the war in World War 2 in South Africa would probably take power, especially after an allied loss that ends all real power that the Unity Party was able to bring to bear.
The Nationalist Party is a republican party so its likely that it will become a Republic in short order given the new world order that emerges after a Nazi victory. The Nationalist party also being white supremacist would find common cause with the Nazis in their somewhat similiar racial doctrines (South Africa just didnt genocide their blacks, rather they displaced them instead)
Either way with the British Empire crashing in flames from a defeat by Hitler, the South Africans would not stick around.