Hey, so, in my TL
The Riders of Hah-nu-nah, the native people of North and South America domesticate horse rather than hunting them into extinction. Leading to a New World that doesn't really fall out of pace with Europe. My first question here is this, what would be the most likely time for contact between the two sides of the world? When is the earliest if different from the most likely.
Thank you,
GM.
Strictly speaking it would be impossible to tell. You are dealing with a new world that will likely have a much larger population than OTL, at least twice as large. This will have significant butterflies all over the place; even the old world would likely be unrecognizable if you take weather patters into consideration. But because that would be impossible to even imagine development, lets keep a butterfly net around the Old World and the Polynesia for the argument's sake.
Assuming that nobody reached the America's before the Vikings did in OTL (very likely the case), the norse will find a more advance society that might have more to offer as trade. An outpost might develop. I think this is likely the earliest possibility for European contact.
If the Viking colony fails as in OTL (lets say they are all killed by natives) then the English and Basques might develop fishing outposts (because this is a more developed New World they might even encounter native fishing vessels around here). But this is just a decade or two earlier than OTL.
Otherwise you need a Portuguese or Spanish explorer running into South America as they try to circumnavigate Africa (once again they might even run into local ships).
Now, there is also the matter of the natives making contact first. If the Pampas develop a large river faring civilization, what is stopping them from trying to sail the sea? If their ships are good enough, and the captain is adventurous enough, one of them is likely to catch a current and land near the cape in Africa.
Or on the other side of the America's following the currents from Peru to Polynesia is much easier than the other way around a Kon-Tiki like scenario might happen had the American's developed better raft/ship building.