I agree that Christianity would no longer be the most popular religion. I guess I exaggerated when I said Western civilization goes bye-bye. Rather, I would think it'd be dominated by Muslims and pagans rather than the native Europeans.
Hey, pagans are native Europeans, too!

The northern European refugees (how ever many there are) might well stop developing. I had intended a Pilgrim-type contact, where there is a peaceful interaction between the native Americans and the settlers. Were the Pilgrims only capable of surviving due to later influxes of people and supplies from Europe, or were they self-sufficient? The settlers don't necessarily have to establish a powerful colony; their primary role in the story is to provide the Americas with low-level contact with the Old World. Even if they regress, they are still technologically centuries ahead of the native Americans. I'm thinking that over the decades and centuries, they could steadily expand outward from their landing site, heading westward and later northward. They won't like to go south too far because of their fair skin.
I assume that they were suffiecent in the sense that they wouldn't die off if cut off from Europe, maybe, as they were very different case from Chesapeake colonial goods colonies (which would most likely perish), but they had to import industrial goods for really long time. Their population was nowhere near enough to preserve knowledge, you need enough population free from other jobs (like, agriculture and basic manufacturing) that makes sizeable "engineering/scientist" population. After two generation, without available iron ore, there might be no one that knows enough to start it up again. And most likely they just blend into Natives.
Now, if they don't start with 100 or something people but few tens of thousands, it probably wouldn't lose that much of technology, and could survive the initial hardship with relatively little loss. There is a problem of moving that large population, i'm sure you see it yourself. Although it's incredibly cool scenario.
That's why i said i think it would take continious long-term contact. Perhaps, you could have Scandinavians not lose link to North America? That way they would absorb knowledge from the south (whatever it is now, Muslim, depopulated...), and NW settlements maintain the much-needed link.
On a side note, it reminds me of that LeGuin book that had slowly dying off off-world population on largely barbarian world. Parts of Rocannon series, i think.
While diseases are exchanged between the NW natives and the OW settlers, so are technology and agriculture. Though the settlers themselves don't make direct contact with the Aztecs or Incas, I figure that their diseases and technology will spread across the Americas much faster, reaching the Aztecs perhaps in a few decades and the Incas (via the Aztecs) a few decades later. I would think that native Americas would see the advantages of firearms over bows and arrows when hunting or warring, and this might accelerate adoption of metalworking, if simply to produce more guns and ammo. I don't know if tradition or fear of the new technology would interfere with this though.
Really, our European refugees firearms are vastly inferior to their own bows

. And require a lot more industrial effort that would be best directed elsewhere. No needs for cannons, as there are no castles, no need for guns as there is benefit for trained archers (all that forests full of game!) and not of large armies. Not to mention it's suddenly a lot more expensive without industrial infrastructure. Bows aren't. Gunpowder itself might not be lost, as i guess it's useful in mining? Metalworking is generally very advantageous, in civilian aplications.
I'm not sure about disease spreading. I'd like to see trade links for that. Granted, i guess Europeans have sea travel technology much, much better than natives, and it enables them to start really good coastal trade. Although it's still a long way from Pennsylvania (that's my guess where the main settlement will be, south enough for warm climate, and north enough for few diseases) to Mexico. Not to mention Peru... It'd take a century or something, IMO.
It's interesting question if/how fast natives are going to adapt. I doubt it would be fear that'd prevent them, rather some structural difficulty, it's not like picking up metalsmithing is a question of stealing blueprints, rather it needs to be teached. So i guess it requires some sort of cooperation or subjugation of refugees coupled with really bright native leader (think Peter the Great of Russia, or Japanese Meiji Emperor)
As for where the settlers might land, if they head straight across the Atlantic, they'll reach Newfoundland. The further north, the colder it will be and thus the harder to set up a successful colony. However, these people are hardy northerners to begin with, so it won't be quite so bad if they're a little far north. Would it be better to have them drift southward and land in New England or the Mid-Atlantic instead? These are desperate folks, simply trying to escape Europe before the catastrophe there wipes them out. So their navigating skills might not be the best, meaning they could land anywhere in the Americas that's suitable for the story. Should I have them leave in advance of the Black Death reaching them so they don't bring it with them to the New World, or would it be better to bring it with so that the native Americans are exposed and can recover from it before the Old World invades in force?
Yes, Newfoundland it is (hey, the name probably stays the same

), but i think their power center will shift southwards gradually. They might be northerners, but i doubt they prefer that to milder climate.
No idea about the plague, though. Was it important part of the disease exchange? And it depends what type of scenario back in Europe we're talking about.
The Aztec Empire was formed in 1325, predating the Black Death. The Incas first appeared in the early 1200s, although their empire wasn't established until 1438. I am assuming that butterflies in Europe won't change the Americas enough to prevent the rise of both civilizations.
Oh, Aztecs that early? Well, 20 years only

I meant Incan Empire, yes. I remembered it being relatively recent in Columbus time. On a side note, Mayas could be in different stage, they had a habit of sinusoidal history.
I thought of an interesting wrinkle I might throw in. If I do go with the 1350 European-American contact, what do you think of the idea that the Europeans to make the contact are Jewish? I'm not certain how many fair-skinned northern European Jews there were back then, but I envisioned a Jewish Canada evolving and got all excited. Somehow, I think Jews would be nicer to and coexist more peacefully with the native Americans than the Christians did in OTL, since Jews have been persecuted as scapegoats for so long. (DISCLAIMER: I seek not to anger anyone over religious matters; personally, I adhere to no religion.) One thing Jews wouldn't introduce to the Americas is pigs, though. If this idea is too unrealistic, let me know.
Heh. Well, Jews didn't have really good record treating their neighbours in ancient times, and i doubt being persecuted actually gives you empathy, on group level at least. On the contrary, sad as it is. Personally i prefer geopolitical to psychological explanations of large scale history. How are you planning Jews to make the trek?
Hmm. I see your point about how much more beneficial to the Americas the other things the OW settlers might bring with them are. I suppose the longer the contact with the settlers, the more information would be shared and the more educated the native Americans would become in European technology, agriculture, politics, etc. (Especially if the settlers remain friendly with the natives, as I'm hoping Jews might.) Incidentally, what sort of new politics might the settlers be bringing with them? Feudalism? The NW already has the concept of monarchy, and as AFAIK, most people of medieval Europe were unfamiliar with Greco-Roman style democracy.
I agree, i just wrote above it requires cooperation (or natives being more powerful than settlers, at least demographically=potentially after some uniting happens).
I see that situation similar to that of Germania between Roman conquest of Gaul and post-Commodus times. Germania got cultural diffusion which basically transformed it from mudhut-living dirt poor area (and still more advanced than NW in quite a few aspects!) to much more sophisticated, populated and powerful place that could threaten Roman Empire. There are two sources, one is Tacitus and another one i don't remember and you can see tremendous change inbetween them (during that timeframe).
So yes, it takes time. I'd give it a century before a trade link is established with Mexico (and i don't want to guess how much time it would take to reach Peru - it could be five centuries, even), another one and you have US natives catching up mostly, and another two-three (that's four-five after the land) with Mexico being mostly upgraded - i gave it 3 centuries because with that distance diffusion is going to be slow. And i'm being generous there.
That's with large scale settlement, that tens of thousands i spoke about before. With Leif Eriksson style, i'd add a century or two before actual NW colony is populous enough that it starts to make an impact.
Which means... assuming Viking colony of 1000 (i don't remember the date), its 1700, and plague settlement it's 1800. Yeah, you need to nerf Spaniards and Europe in general, or rewind to Phoenicians or something
About the politics... it's hard to pinpoint and i might be wrong anyway. It's not as easily grasped, as say, metalworking, and i don't have that much knowledge about it, but you know, Italians had insurance companies by 1500 and banks even earlier (ok, that's not Scandinavians... throw some Flemish or proto-Hanseatic trader folk into the refugee fleet then), schooling institutions like universities or monasteries, guild system... not politics, although politics too, since you got more power distribution in European population than in Inca (and, i guess, in Mexico too), and i think it's good for sophistication and progress. Granted, NA natives tribal stage could be similar, but... oh, it's really complicated
Say, it's too decentralized and not powerful enough state. And while totalitarianism Egypt style is powerful, it's bad for different reasons. But yeah, that's not scientific knowledge, but my (un)educated guess.
I like the Mongol idea you have there, but how exactly could Song China hold off the Mongols? Good old Genghis died before the Mongols reached the Song; instead it was Kublai who conquered them in 1279. Do you suppose the death of Kublai Khan would be enough to prevent the Mongols from taking over the Song, or would the Song had fallen to the apparently superior might of the Mongols anyway? Even if Kublai died, wouldn't the Mongols still want to get rid of Song China first instead of Europe, since the Song are right in their backyard?
Damn, i have
no idea. I just don't know enough about internal situation of Song China to even guess. Including the reasons of it's collapse.
But, one thing, i wasn't talking about Southern Song, but Northern Song. Which means, there is POD regarding Jurchens. Or, it's Jurchens that defeat Mongols. Anyway, i guess it's worth exploring.
(on a side note i agree that whole Eurasia under one rule is unconvenient in long run)
Another of the many things that concern me about this TL is when the Chinese stopped Zheng He's overseas exploration and focused on internal affairs, officially cutting off contact with the outside world for a long while. I'm not certain, but I think this isolationism slowed the development of Chinese civilization compared to the Europeans. Do you suppose that even with this isolationism occurring as in OTL, the absence of the Europeans and Americans who cooperated to castrate China would permit China to remain a superpower to the present day?
Oh, that depends if there are Muslims who do the same 
And i'm not sure if it was outside influence that actually prevented Chinese from anything, rather than internal factors. And btw, wasn't Zheng really a explorer and not a diplomat? Again, it's not like i know that much about that, but i didn't have that impression it was sudden change and one decision thing that transformed China... but perhaps you should start another thread about that, as i doubt anyone is still reading my huge post 
Which leads us to...
In any case, pushing the POD back in time from 1347 and having the Song dynasty survive could prevent the isolationism and give China an even better head start. Hopefully not too much of a boost, though. A Chinese-dominated world would be very interesting, but I'm looking for more of a balance between China and the Muslim civilizations.
I just realized it's another problem. Look at the recent Baghdad WI thread, it has some interesting things regarding Muslims in that time. It might require a POD - although not necessairly, as you're reshaping a lot of things already, so while southern Muslim world stays as in OTL, northern (ie: area formerly known as Christian Europe), obviously does not. And might lead the progress, even though slower, due to ATL events, than OTL. Which is what you want, anyway.
But yeah, if you give China too much of a boost, we're all fucked 
Having the Mongols conquer Europe seems like the obvious solution to my problem, but for some reason it doesn't sit right with me. If Ogedei had not kicked the bucket in 1241, it is indeed likely that the Mongols would have taken Europe? Check out the assessment by antisocrates from another thread:
Yes, i'm not all that enthusiastic about Mongols warring in western Europe with only slight changes. That's why i suggested something so Mongols have basically no choice than turn west in force after Northern Song or Jin defeat them.
On a side note, keeping China split into Southern Song and Jin is quite interesting too. And doable, i think.