I've thought plenty of this for my setting, and I've come to the conclusion that this completely changes North American geopolitics, because Scandinavia will be a lesser greater power than OTL France, and also lack the Catholicism prominent in New France. Either way, I'd expect the St. Lawrence to be the main corridor of cities and settlements and similar decline of the Maritimes. I suspect it will be settled from OTL Greenland, Labrador, and Newfoundland as part of the Danish crown asserting Norway's claim to those lands. This may extend into Hudson Bay because Baffin Island (Helluland) is also a claim of the Norwegian crown. But eventually the claim will move south since the St. Lawrence has the richest and most numerous native nations (well, before the destruction of the Laurentian Iroquoians). Either way, the St. Lawrence will be Norse ITTL, and given OP's constraints, they will take most all of New England and upstate New York too.
I do not think they will reach all the way to the Pacific because OTL British Columbia was pretty much just British, American, and the occasional Russian and Spanish ship. There is little ability to colonise there, and it is very likely a separate British colony can arise there and probably protect itself against American influence given New England was the source of American ships and New England will be Vinlandic TTL. But I believe they could easily seize the Prairies, will compete for the Upper Midwest, and will have a similar mindset to build a railroad through the Canadian Shield.
This will be a darker version of Canada, where they will have the same OTL Canadian trait of paranoia over a more powerful southern neighbour but expressed much differently because Scandinavia can never be as dominant as 19th century Britain. This will likely play into a militarised stance akin to at minimum what Sweden did in the Cold War--neutral, but armed. It will be industrialised with a strong fleet on the Great Lakes and St. Lawrence and likely all sorts of coastal defense ships. Expect numerous fortresses in all places, which will be a stark contrast from OTL New England with its quaint mill towns, ice harvesting, and timber harvesting.
But like Canada, they will very likely enjoy hockey though, so some things remain the same. They will also be influenced by the Americans and have a strong baseball culture (so something like pesäpallo will fuse into baseball) and maybe American football.
I suspect it would be Danish for the most part, being the most populated nation, and having less in the way than Swede (i.e not having to go through Denmark-Norway.) That said, it would also be less populated than the English colonies
It isn't so much population, but proportions, so considering the settlement patterns and history regarding the Forest Finns, I suspect any colonial minded administrator looking for settlers would turn to Finland. In New Sweden for instance, the majority of the small population came from Finland, and this is the period where Forest Finns carved out settlements in northern Finland, Sweden, and Norway. If we assume it's Kalmar, than this also helps in getting the Swedish crown involved in the colony instead of their own interests.
I would assume the Finnish language in Vinland (and I guarantee the name "Vinland" would remain in use for at least some part of this colony, if not the entirety, or as a post-independence name from romantic nationalism) would be at least as common as Canadian Gaelic, and at most optimistic, as common as Canadian French (assuming nationalist cultural-linguistic revival on both sides of the Atlantic, like maybe Finland is separated into its own kingdom/grand duchy to counter something like the OTL Russian plot to do so). I've always imagined Finnish culture/language mostly concentrated in the OTL Maritimes and fringes of Quebec.
And while northern Finland/Sweden/Norway was OTL the target for Finnish settlers, much of this area, especially OTL New England, has a better climate and land than that area (if much more dangerous due to Indian wars). Finnish immigrants to the US and Canada settled similar areas to their homeland (northern Michigan, Minnesota, northern New England, Oregon/Washington).
The second major group of settlers would probably be Icelanders and Norwegians, since their lands often suffered famine, or for the former, volcanic eruptions. The Laki eruption in the late 18th century would lead to thousands of settlers.
If they're not being picky, then there is no real limit to how populated it can be. If they convince Germans to go there and adapt somewhat to Scandinavian culture then it could be pretty big and it may even eclipse the home nation in terms of people and resources. Convincing others to go there, would probably save the home nation from becoming too sparse.
That's true too. A united Scandinavia would be very active in northern Germany and thus have yet another base of settlers.