Tolkein never did reveal much about what Mordor was like after Sauron, save that after Sauron's death, Elassar granted Mordor to Sauron's freed slaves. Tolkein also never went into much detail about Sauron's slaves either. So since we know very little about these elements, the best we can do is guess and make theories. Here are my theories.
The Kingdom of Nurn was the successor nation to Mordor. Slaves who had only recently been enslaved were allowed to return home, but those who had lived their whole lives in the shadows of Mordor chose to stay. I'd imagine that Sauron's slaves came from every human nation in Middle-Earth, or had ancestry going back to the human nations. I also assume that they probably intermarried and mated with eachother, creating a new ethnicity of Men, Nurneans. In other words, they have resemblance to the multi skinned people of the Jacksonverse Dale and Laketown.
In terms of culture, I doubt any Nurneans would have any familiarity with the cultures of where their ancestors came from, so I'd imagine that Nurnean culture would be a new arrival in Middle-Earth. Since these were freed slaves who were probably treated in apalling conditions, they probably had a lot of relief aid from the Reunited Kingdom and Rohan in order to get at least medicine, education, etc. Some elements from Sauron's Mordor may have stayed though. Sauron would still have needed roads in order for trade to come to and from Nurn and Gorgoroth, so those probably stayed in order for trade to continue. Abandoned Fortresses were probably rebuilt into cities, towns, or villages.
In terms of actual culture culture, I'd imagine that Nurneans were a friendly, but somber people. All their lives, they have known nothing but darkness and evil. Though they are not evil themselves, they are still creatures of the dark. It's taken them longer to get used to sunlight being more common. Most Nurneans were likely introverted or socially awkward, maybe even similar to modern day people who like the gothic style. In terms of architecture, I'd imagine that since these ex-slaves were forced to build dark and gothic buildings, Nurnean architecture would also look dark and gothic, but would look at least a bit more welcoming. After all, this is all they've known, they've just modified it to be something good.
I'd imagine their military would have resemblance to that of the Black Numenroians, with elements of Rhunic, Haradric, and Khanish armor as well. Their armor is dark, black, grey, navy blue, maybe even some crimson red. This makes them look frightening on the battlefield, and thair fighting style was probably an amalgam of various mannish fighting styles. Their weapons are probaly similar to the weapons of that of Easterlings, Haradrim, Variags, Black Numenorians. Their not pretty and look terrifying, but they do their job. And since horses were rare in Mordor, I'd imagine that either the army was all infantry, or the cavalry rode Wargs and other dark creatures found in Nurn.
In terms of poltics, I assume that like most nations of Men, Nurn is a monarchy. The first King or Queen was probably a senior slave with leadership skills. After all, in real life, slaves had leaders among them, even if the leaders liked or their masters. But that's as far as I can predict in terms of politics. I can just as easily see the heir to the throne being chosen by birth or by merit, since that's how the first King or Queen rose to power. I also don't know if this would happen in Tolkein's media, but if I could have my way, I'd also establish that women can become ruling queen and have more say then any other human nation.
So that's what I can make out of the vagueness of what Toklein wrote. What about you?