We don't really know.
Tolkien seems to have intimated that Sauron kept a population of Black Númenóreans as his elite Lords over Harad, Rhûn, and Núrnen (South Mordor, where all of the food was grown).
And, if you look at the Logistics, from 2899, when Ar-Adûnakhôr takes the throne, the political divisions within Númenóre are finally well-established, and the lands of Middle-earth, formerly allied to Númenóre officially become subjugated vassal-states.
This means that Tens of Thousands of Númenóreans, affiliated with the King's Men (rather than the Elendili) were given positions of Power within Middle-earth.
Specifically in Gondor and Harad (Umbar, which was then the seat of Númenórean power in Middle-earth - the vice-regent ruled from Umbar), where the vast majority of Númenórean settlements were established.
Then, when Ar-Pharazôn captures Sauron, he increased the subjugation of Middle-earth tremendously, leaving tens of thousands more Númenóreans in Middle-earth.
Now, there would have been a reduction of these, but remember that the Eledili, under Elendil's father, Amanda, had been working to place the Elendili in Middle-earth as a safety against the number of King's Men who were there, so that when Ar-Pharazôn's eventual heresy culminated, that as many Noble and True Númenóreans could be saved as possible.
So, even with Ar-Pharazon drawing down the manpower in Middle-earth for his assault on Valinorë, the Elf-friends would have still remained outnumbered.
After the Akkalabêth, many of these King's Men would have remained loyal to Ar-Pharazôn, and thus to Sauron, but the vast majority would likely have remained true to Númenóre itself, and thus recognized Elendil as the rightful heir to the throne of Númenóre (or, the New Kingdoms of Westernesse in Middle-earth established by Elendil), even if they personally loathed him (duty was still considered to be a quality they valued, even if their duty was to an office, rather than to the person holding that office).
Given the numbers of Black Númenóreans who would have survived The Llast Alliance, there was a sufficient population of them to have been viable, genetically (even though Tolkien would not have considered this, exactly), and with Sauron's aid, could have unnaturally extended their life-spans (so that they lived to 500 or 1000 so years of age, even if they were decrepit and infirm past 200).
It is likely this was the purpose of some of the many "Lesser Rings of Power" that Tolkien mentions via Gandalf and in Letters, and was likely a prerequisite for many of practitioners of the occult religions Tolkien mentions being popular in the East and South.
So. . .
Black Númenóreans likely existed well into the Fourth Age, not just into the Third.
MB
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