I would imagine it would be a similar situation as other examples where two countries (or two different areas/territories) share a name.
So we had or have:
1. German Democratic Republic and Federal Republic of Germany ---> East Germany and West Germany ---> East Germans and West Germans.
2. Democratic Republic of Vietnam and Republic of Vietnam ----> North Vietnam and South Vietnam ---> North Vietnamese and South Vietnamese.
3. Democratic People's Republic of Korea and the Republic of Korea ----> North Korea and South Korea ----> North Koreans and South Koreans.
4. People's Democratic Republic of Yemen and the Yemen Arab Republic -----> South Yemen and North Yemen ----> South Yemenis and North Yemenis.
5. Republic of the Congo (previously People's Republic of the Congo) and the Democratic Republic of Congo ----> Congo-Brazzaville and Congo-Kinshasa ----> Congolese.
6. Republic of Cyprus and Turkish Republic of North Cyprus ----> Cyprus and Northern Cyprus ----> Cypriots and Turkish Cypriots (or more rarely Northern Cypriots).
7. Republic of Ireland and Northern Ireland ----> Ireland and Northern Ireland ----> Irish and Northern Irish (the term "Southern Ireland" was used briefly but fell out of favour very quickly for good reason).
8. The Independent State of Samoa (formerly Western Samoa) and American Samoa ----> Samoa and American Samoa -----> Samoans (formerly Western Samoans) and American Samoans.
9. North Macedonia (formerly Macedonia) and Macedonia (Greece) -----> North Macedonia and Macedonia ----> North Macedonians and Macedonians (formerly just Macedonians for both).
10. People's Republic of China and the Republic of China ----> China and Taiwan ----> Chinese and Taiwanese (but Chinese for both in official terminology).
Assuming there was a southern victory and recognition of southern secession by the USA, then unlike in say the Cyprus case, the CSA won't be an unrecognized near-pariah statelet wherein the USA would continue to be considered as the only "true" America and thus only US citizens being called "Americans". It would be unlike the Irish or Samoan situations in that both would be independent and therefore equal entities.
It is thus more likely (in English) to be similar to the cases with Yemen, the Congos, Germany and Vietnam.
So:
United States of America and Confederate States of America ----> USA and CSA ----> Union Americans and Confederate Americans.
This might be shortened further into just "Americans" and "Confederates" in some speech especially when by the context it is quite clear that "American" in this case is being used to refer to only citizens of the United States (for example in a western context when people say "Korean" they almost invariable mean South Korea. You don't expect someone to be referring to North Korea when discussing baseball, Korean dramas or Korean pop music). However "Americans" is also likely to be used to refer to both Confederate Americans and Union Americans much as how "Germans" was used to refer to West and East Germans or "Yemenis" was used to refer to North and South Yemenis and how "Congolese" refers to persons from both Congos.
And much like in those cases, locals and politicians within the "two Americas" would likely refer to themselves as "Americans" but refer to persons from the other country as something else. So West Germans called East Germany "die Zone" or "die Ostzone" (the eastzone), while West Germany "Bundesrepublik" (Federal Republic) or as Westdeutschland (West Germany) by East Germans. And of course there are the terms Ossi and Wessi. In Korea I gather that South Koreans and North Koreans refer to their countries by two different names entirely (Joseon and Hanguk) but also use informal names for themselves and the other side (like north country and south country or northern side and southern side). So in the United States, I would expect that in common parlance or in political speeches and campaign rallies the citizens of the United States would be referred to as Americans while persons from the CSA would be called "Confederate Americans" (in polite terms) "Confederates" (in slightly more informal terms but also derisory terms), "Rebs" or "Dixies" (in derisory terms), "Southerners" (in very informal terms but also derisory terms) or "Southrons" (in very informal and likely very derisory terms). Meanwhile down south in the CSA locals and politicians would call themselves "Americans" in everyday speech and political speeches and campaign rallies, but refer to people from the USA as "Union Americans" (in polite terms) "Unioners" (in slightly more informal terms but also derisory terms), "Yankees" (in derisory terms), or "Northerners" (in very informal terms but also derisory terms) or "Northrons" (in very informal and likely very derisory terms).