The most likely scenario would be a joint emporium, soimething like the steelyard or the tyske brygge. That would have to be set up in an area that already had trade relations with the Hansa, so it would either be in the context of someone else's colonial undertaking (maybe Burgundian, for the hell of it) or in cooperation with a powerful native state.
If no such partner exists, but the Hansa cities still feel the need to trade, they may simply set up seasonal camps, Newfoundland-style, and invite natives to barter. Some people might stay behind when the convoy leaves, but they won't be so much real colonists as woodsmen, securing the supply of trade goods by tapping into native treade networks.
I can't see any lasting impact on the American side because the Hanseatic cities, even if they retained their cohesion, are poorly placed to exert that kind of influence. They do not have people to export - most of them had negative population growth and imported migrants - and they are averse to fighting wars, which means they lack the military power to play for big prizes "beyond the line". A surviving Hanseatic league would most likely have a small part of the American trade, mostly using its experience in Arctic shipping to harvest natural resources like cod, whale and walrus oil, seabird fat, and traded furs on the Canadian coast. Until the big players pushed it out, if they do ITTL.