The various Southron and Easterling factions were described as having their own struggles against one another, and as such, would probably not appear together without a unifying figure of some sort.
Just as Khamûl was supposed to be the ruler of the Easterling Empire (nominally centered near the Sea of Rhûn), it is highly likely that the other "Fallen Realms¹".
Harad, for instance, was described in The History of Middle-earth, book IV: The Shaping of Middle-earth as being composed of many different Tribal states, with some being dominant (such as the lands near Umbar being controlled by the large population of Black Númenóreans who had settled there. Supposedly, there was another large settlement of Black Númenóreans further to the south (but not nearly so large as that at Umbar).
And Far-Harad was supposed to be even more tribally-balkanized.
The Khandirim were a grouping of two peoples: the native Variags, and then an immigrant population forcibly re-located there after the Battle of the Camp, when the Wainriders were defeated after their last major incursion into Rhovanion, Calenardhon, and Gondor.
And the Easterlings were said to be a large Empire or Confederation of states in the East of Rhûn, ruled over by Khamûl.
It is likely that other Nazgûl were either taken from the rulers of the other Realms, or they eventually established suzerainty over these areas after the rise of Sauron.
This would have given a greater ability to cooperate between the different realms.
But absent a powerful guiding figure, it is likely that the "Fallen Realms" would have had a hard time maintaining alliances for long periods of time, as one of the key themes Tolkien discusses is the greed, lust for power, and cowardice of those under the Shadow.
This would lead to situations where one realm might do something during a campaign that placed another realm in a dangerous position, the effects of which would see the former realm profit at the expense of the latter realm.
Such actions would tend to breed antipathy among these sorts of people.
Another thing mentioned by Tolkien, though, is that the "Fallen Realms" were populated by peoples who had been lied to by their leaders and either Sauron, or his emissaries, such that the majority of the populations themselves were not "bad." But rather that they had been manipulated by fear, and oppressed by their leaders in such a way that reinforced behaviors among the population that rewarded "bad behavior" (from the standpoint of the Realms of the West, and the Edain).
But from a scenario point of view, there are many periods where you see the various factions seemingly cooperating for periods of several hundred years at a time. Although it is likely that the "cooperation" was pretty tenuous and strained, as Tolkien only describes the major battles that produced a turning point in these wars (wars which lasted for, as I have mentioned, up to two-hundred years at a time). So it is likely that there were major battles (or larger skirmishes, at the very least) that took place almost yearly during this time where you would see various combinations of factions involved.
But this would probably be limited to combinations of the Easterlings and the Ancestors of the Khandirim, and combinations of the Variags of Khand and the Haradrim/Far-Haradrim before the period of the Battle of the Camp. And you would see combinations of the Khandirim (both Variag and Wainrider) and Haradrim/Far-Haradrim after said battle. After The Battle of the Camp it would have been difficult to get Easterlings and any Southron together for a battle until the years right after the beginning of the War of the Ring as Ithilien was still held reasonably well by Gondor up till that point.
¹ The term "Fallen Realms" is really a poor choice of words for these countries, nations, and states. It seems to imply that they once were aligned with the "right side" in Middle-earth (i.e. had not strayed or fallen under the Shadow), but the reality is that all mankind at one point had fallen under the sway of Morgoth, and that the Three Houses of the Edain, and the associated branches of these people who did not travel into Beleriand were the exception rather than the rule.
So the term Fallen Realms is really redundant. The Good Guys should really be the one's singled out as being different than the typical realms of men.
But I understand that GW is trying to market a game, and that subtleties of this sort get lost in the need to communicate a meme designed to evoke a certain prototype.
