So who do have can settle the New World?
Greater Norsca: This nation is made up of what’s left of the Norsca combined with what’s left of the former Empire province of Nordland.
Kislev: Kislev now controls all of what was original Kislev and most of the former Empire province of Ostland. Maybe a little bit of expansion eastward into lands formerly controlled by Chaos friendly tribes.
Estalia: Once a puppet state of the Empire of Man, Estalia is now fully independent and looks exactly like it did during the Warhammer World That Was. Well not exactly, maybe about 15% of its land sank into the sea.
Tilea: Once outright conquered by the Empire of Man, Tilea is now fully independent and looks almost exactly like it did in the Warhammer World That Was. Minus about 5% of the land falling the sea.
Westland: This is made up of the Empire’s former southwest corner. Once landlocked, the terraforming that followed the Demon Wars now gives them reason sea access.
New Ulthuan: After the dust settles, Naggaroth remains almost the sole home of all living elves. Since most of the new rulers hail from Ulthuan, they renamed the land New Ulthuan.
The Goblin Lands: The land that used to be the Border Princes has a lot of coastline, so the Orcs and Goblins can sail to the New World without a lot of fuss.
Who wins the race to grab as much of the New World as possible? Kislev wins. Estalian takes a close second. Greater Norsca and the Goblin Lands fight it out for third place. There are lot more Goblins in the New World than Norsemen, but the Norsemen in the New World are a unified block and the greenskins are not.
The other groups do make it to the New World, but they are so isolated that they either set up isolated trading posts or they are force to bend the knee to one of the main powers.
New Ulthuan had a good head start since they are pretty much adjacent to the New World, but it takes them well over two centuries to stabilize as the former Dark Elves and High Elves have to get used to the fact that their former worst enemies are now their countrymen, so their head start vanishes and then some. They also replenish their numbers slower than humans. Some elves can visit and trade with the New World, but large scale colonization is out of the question.
Greater Norsca has the strongest seafaring culture of all the newly emerged human nations. Their explorers are the first to map the New World and they are the first to set up colonies. Unfortunately, all the native people and the supernatural monster threats have no one else to fight with, so most of the early Norse colonies fail. The Norse cannot keep up a sustained effort at mass colonization because their base population is small relative to the other human nations.
Estalian has a seafaring tradition second only to Norsca. They have a greater population to draw on. For almost the entirety of recorded history, their demographics were unchanged, so they were not slowed down by political instability. They are the first Old World power to make major inroads into the New World.
Brettonia could have given Estalia a run for their money, but they had a problem. They were really happy to throw off their Empire oppressors and reclaim independence, but internal dissentions were great. Traditional Brettonian culture has utter contempt for their peasantry. After centuries of enjoying the relatively decent treatment they received under the Empire of Man, the Brettonian lower classes overthrew their nobility and created a Republic. Not having a lot of strong leaders, the Republic failed, only to be replaced by a new monarchy that looked a lot like the various breakaway states from the Empire. By the time the dust settled, all the other naval powers had a good head start on the Brettonians when it came to establishing overseas colonies.
Tilea was now more unified than it had ever been, but they were the only naval power that was bordering the Goblin Lands. That means their navy had to focus on defending their coast from greenskin pirates. They couldn’t afford to send massive expeditions out west.
The greenskins of the Old World Goblin Lands are not especially interested in colonizing the New World. They don’t have the mental discipline to make an organized colonization, but they are brave and foolhardy enough to make unorganized colonization attempts. A few of the New World greenskin enclaves survive if not thrive and because greenskins reproduce very quickly once they get in the New World, they are very hard to dislodge.
Westland used to be landlocked, now they weren’t. They did not have a preexisting naval tradition so they had to build one up from scratch. Their coastline used to be dry land, so most of their waters were pretty shallow and difficult for all but the most experienced sailors to manage. By the time they had a large number of experienced sailors, most of the New World was settled. But Westland does have a very large population. A lot of Westlanders chose to immigrate to the New World, but they had to find a place in other nations’ colonies. Kislev had the most similar language and culture both being cultural heirs to the Empire of Man. In fact the wave of Westland immigrants eventually paved the way for a trickle of immigrants from West Landlockia to follow in their footsteps.
Kislev emerged as the first early rival to Estalian dominance of the New World. Once the Kislevite colonies swelled with Westland immigrants, the Kislevite colonies had the human fodder they needed to hold the decisive advantage in any New World War with the Estalians.