For the first month after the death of King Viserys, his Council managed to hold the Kingdoms together but then contenders decided to seize control. Among the first were Lord Borros Baratheon and Lord Hobart Hightower. Borros claimed that since the Baratheon's were a bastard line of House Targaryen, he was the logical successor of his distant cousin. He had quite a large number of supporters.
Hobart, however, claimed that the charges of treason which his brother and niece had been sentenced to death on had been falsified and that they were manufactured by their detractors who had also poisoned his niece which prevented her from birthing many Hightower Princes. He, therefore, claimed that it was his right to become the new monarch since, had it not been for the crimes of others, he would be the closest living kin to the young king. Plus, they were the longtime defenders of the one true faith.
Hobart also had his supporters and among them were all of the Faith and Maesters.
To make matters worse, of the paramount houses, only Houses Baratheon, Greyjoy and Stark still were viable with heirs and the only kingdoms who's nobility hadn't suffered too greatly by Viserys' actions with their wives and daughters were the Iron Islands and the North, both of whom rarely came to court and were usually ignored by the other kingdoms.
Hobart and Borros lead their forces to a battle at Harrenhall were ultimately both were killed along with Hobart's three sons, none of whom had yet married. Borros only had three daughters, all of whom had been among Viserys' wives.
Following this battle, others from more minor houses tried to seize control of the throne but the longest anyone held it was less than a month before being removed. During this era, the North sealed the Neck and declared their independence once again. King Cregan Stark was the first King of the New era of the North.
Less than five years later, the Reach and the Stormlands had been conquered by Dorne.
Then, less than a decade later, Vikings came to raid and settle. They merged with the Ironborns and ended up with control of the Westerlands and Riverlands.
As for the Vale, it fell into chaos following the death of Lady Jenyne Arryn and remained disputed for decades to come. However, with time, the oldest child of the oldest child of Lady Baela Targaryen, the firstborn child of Prince Daemon and Lady Laena, chose to return to Westeros following years of her family's conquest of Essos and their resettlement of the now broken but calm islands of the former Valyrian Peninsula. She was the first dragon in Westeros in a century and she wound up only claiming the Vale and Crownlands for herself and her descendants.
People would ask why she didn't conquer the entirely of Westeros in which she would reply how she wanted to honor her Great Great Aunt Queen Aemma Arryn by ensuring that the only two homes she'd ever had were protected, that way, should a descendant of her daughter, Countess Nyra ever return, there would be a safe place for them.
It was also about this time that the Faith of the Seven finally died completely. Greater Dorne's official religion was R'hllor and they banned the Faith. Austrlond was the name given to the combined region of the Iron Islands, Westerlands and Riverlands by their Norse conquerors and they would continue following the traditional Norse faith into the modern era. The Faith was eliminated from their rather quickly as well. When Lady Alysa Fyre of House Targaryen retook the Vale and Crownlands, she immediately banned the Faith, ordered all Septs destroyed and imported religious leaders from all other faiths, including Celt, Norse, and Christian. Only the Faith of the Seven was banned from her lands. The North had never permitted that religion so it was never eliminated from there.
Thus Westeros became four nations which it never again changed from though Greater Dorne would eventually become known as merely Dorne and Lady Alysa's lands would remain a principality of New Valyria.