Well if we just 'remove' the Thirteen Colonies and that's it, France would obviously be the one to step in. At least north of Mexico, they had the most settled colony in French Canada. No English colonies, means the French hold onto it.
However that ignores the butterflies on whatever would hinder or deny English colonization entirely. My thought is the War of Roses goes/end differently or the reign of Henry VIII goes even worse. The civil wars of England over religion or succession are much worse, Henry VIII either never establishes the Royal Navy, or too much funds for the RN are spent waging civil wars. England never becomes the naval power, etc. So my thought would be a POD in the early 1500's. Now I'm just doing a quick scenario I thought up in twenty minutes. I'm making huge generalizations.
France would almost certainly still dominate. It would want the Newfoundland fisheries, and that almost guarantees the St. Lawrence. So French Canada is almost certain, and it was the colony in NA that got the most historical French settlement. There's also no one else as well positioned as England to battle France. No one else can put so much of their faith in their navy as England could and did, not the Dutch, not Denmark, not Spain. This means outright defeating France in its golden age is probably not going to happen. However France wasn't exactly a country that sent lots of its populace to its settler colonies, so it's not like France automatically gets everything.
Both Spain and Portugal could be options for filling the vacuum, but I don't think so. Portugal will suffer under the Iberian Union and has Brazil to focus on, and Spain just has too much land already. The Dutch are a natural to step in. I don't think Scotland or Ireland could step up with a weakened England, they'll likely be so focused on the power struggle in Britain. They're also suffering under that which most states are. As long as they need to worry about attacks from their nearby neighbors, they can't put as much focus on the navy as England was once it achieved dominance. Sweden I assume is out. Never a colonial power, New Sweden only came into being at the height of its power in the Thirty Years' War. Denmark I could see though. It's great problem was that it was always focused east against Sweden due to historical issues, and to the west it was facing England and the Dutch. If England is crippled and looking inwards too much to colonize, I could see Denmark-Norway with a greater sphere of influence to the west. The Dutch are still there, but less likely to clash.
So the Dutch will probably be the immediate second to France, during the Dutch Golden Age. It's rich and urban with a powerful navy, and there will be no Anglo-Dutch Wars to cut it down here. New Netherlands covered much from Cape Cod to the Delmarva Peninsula. With no English Colonies in New England, I assume New Netherlands to spread to that area. Maybe contested with the French. However the population wasn't too great. Less than 10,000 by 1664, historically. While a Dutch dominance might lead to more settlement, it probably wouldn't match England OTL.
I'd say the Danish would move in. If we assume New Sweden even comes into being, I could see Sweden eagerly trading it to Danish control as a concession for the far more valuable land contested with Denmark in Scandinavia. A western focused Denmark would probably also have started other areas of settlement. New England is possible, creating a conflict with the French and Dutch. However without Virginia being established, the land further south of there would be open. One important note is that with Holstein under the Danish Crown, I could see Denmark's colonies seeing an influx of German settlers during an alt-Thirty Years' War.
So I see a French-Dutch-Danish contest, with the first two having the advantage at first. France would probably be able to extend its control of the Great Lakes. This almost certainly leads to a French super Canada that would probably then extend west to probably include all of Oregon Country. This would greatly affect the contested zone of New England east of the Great Lakes. Probably contested between the Dutch and Danish previously, with its westward expansion cut off it would probably become a lesser arena compared to the southern areas which still can spread west. So either they would simply partition this area, or one would triumph and the other would accept it to focus on battles elsewhere.
So the main colonial battle between the Dutch and Danish would be based around OTL Maryland and south of that area. I'd actually go and say Denmark would have an advantage. One, if Denmark could draw German colonists it would have a steadier flow of settlers. Primarily though, the Dutch are unfortunate enough to be close to France. Any colonial conflicts would lead to French armies marching for the Netherlands, and even if peace is maintained, France would almost certainly go for the Spanish Netherlands at some point. France gaining those lands is unacceptable for the Dutch. The Dutch basically have the riches and power to be a colonial power, but its population and position on Europe make that a brittle power which means France would probably cripple it in a similar manner as the English IOTL. After that, it is simpler smarter for the Dutch to focus on their exploitation colonies over their settler ones for profit and trade. New Netherlands could potentially be engulfed like OTL, but I'd guess not as assuming the Dutch stay in the game till 1700 when the WoSS came about to ensure the Spanish Netherlands became a conflict, with my assumption of British settlers coming to New Netherlands, it would have a population enough to survive if not expand.
So Denmark-Norway probably has much of New England, which are cut off from expansion by French Canada, and probably much of what IOTL constituted the southern states. These two would be separated by New Netherlands. It would be populated by Danish, Norwegians, Germans, and probably Swedish, Finns, and Brits too. I'd honestly assume these lands would break away very similar to the ARW as Denmark isn't as powerful as Great Britain was. Although maybe as Denmark doesn't have the colonial Empire elsewhere as Great Britian did, it wouldn't be so heavy handed. It would only care about securing its wildly successful North American colonies, even if it means giving far more self governance. I could even see these lands balkanizing if the ethnic groups haven't quite blended well enough.
France would still have Louisiana, but without major changes in French settlement there I think it would inevitably be taken. I don't think that changes would happen, it's possible that without a major rival in North America that France would devote even less resources to their colonies there, possible meaning that French power might protect Canada at first they might find that more rapidly growing weaker colonies surpassed them by 1850~. So Louisiana probably is taken, although I could see with a weaker Danish alt-US that the French could team up with Spanish Florida to keep the entire Gulf Coast. So they'd only lose those parts of Louisiana blocking Danish westwards expansion.
Spain's land in Texas and California I could see being major areas of conflict. Spain would still have its OTL problems of having too much underpopulated lands, which would always be a vulnerable to a westwards expansion from the eastern seaboard. I'd assume this Danish alt-US would fight for a western coast, and with French Canada here well positioned to take Oregon Country that means they'd have to take some of California. Spain would probably have to compromise at some point.
Although another issue is if England in this TL also doesn't participate in the colonizing spice or sugar islands. The Caribbean would be an entirely new theater to further complicate the issue, as sugar islands were massive sources of revenue. I've already summarized a scenario without enough detail to expand it though. So even if this is a quick scenario thought up, I hope it isn't the most unrealistic scenario possible.