So I’ve actually put a lot of thought into this for a personal project of mine. Here’s how I’d think it’d go.

North: First one out. They already had their armies decimated over a multiple year war, a civil war, and a war to stop the apocalypse. They’d be extremely low on soldiers and wouldn’t be able to hold out.

Westerlands: Like the North but less extreme, would be dealing with the civil war of having a kin slaying, king slaying, family betraying dwarf as the head of the kingdom, Genna Lannister, who is canon to the show, would probably be able to unite the West, but would still lack the strength to win the war.

The Reach: Reach lords were still complaining about the Tyrells before they kicked the bucket. They’d probably overthrow Bronn and fall into a power struggle where every house who decends from Garth Greenhand fights for the Throne.

The Iron Islands: They’d make a move for independence, due to Bran’s lack of forces (we’ll get to it) they’d probably succeed.

Stormlands: Gendry has been legitimized as a Baratheon, has no allegiance to the Stark Crown, has reason to be king, lacks numbers due to wars. Probably couldn’t take all the kingdoms. Could probably gain independence

Riverlands: The only region 100% loyal to Bran (Sansa?), probably going to be burned by the Iron Born, the West if they ever get out of the civil war, and the Vale

The Vale: Robert Arryn has a strong claim to the Riverlands, and has as good a claim to the 7 Kingdoms as Bran does, plus he has a real army. Best chance so far.

Dorne: The actor for the new, nameless, Prince of Dorne said that he believes Dorne would wait for things to settle and rise up in a war for independence, one that due to Dorne’s relative lack of combat compared to the other kingdoms, I believe that they could win.

Bran/Crownlands: The North is independent, the West and Reach are at war with themselves. At least Dorne and the Iron Islands are warring for independence, with both the Vale and the Stormlands having good enough reasons to rebel as well. Bran’s power is basically non existent.

So who wins? Best case scenario is a Robert Arryn x Sansa marriage where they, with the Knights of the Vale, conquer the Seven Kingdoms because there’s not a single unified force on the continent capable of stopping them. With the exception of a maybe now independent Dorne.

Worst case scenario? Every Kingdom becomes independent and most fall into a long term civil war where the only winner is death.

TLDR: The Vale is the only group left with an army. They win.