Anywhere will do
I asked a very similar question before. I had a fixed point of origin (Haiti) and asked for the most likely landing spot. But your question, like mine, does not have an answer. Let's examine the evidence, and take my map of Columbus's voyages from the New World to the Old:

The different arrows represent prevailing winds (broad grey) and ocean currents (narrower blue) respectively. Columbus' return trips are imposed on top. And what you find is that he barely followed any of these 'logical routes'.
From this evidence I have surmised that prevailing winds/currents make a particular route easier, but not any of the others impossible. Your ship heads in the direction you steer it in. Going directly against the wind won't work for a sailing ship, but if you decide a course in that direction, what you do is you alternately go to either sides of that course, so you move in a zigzag pattern and end up going in that direction. That is until the wind changes again; prevailing does not mean exclusive.

The better the ship, the sharper you can go against the wind, but it's all but requisite for a sailing vessel capable of crossing the Atlantic in both directions to be capable of doing more than just follow where the wind leads it.
So view this from the perspective of your Native American civilisation. Maybe it's Aztecs, maybe it's the Guanahatabey. Whatever their reasoning for crossing the Atlantic; they would have had something they expected to find. And that something would have (theorised) coordinates. So they would go wherever they wished to go, and if that imagined destination happened to be at the place of real-world Europe, then that's where they land.
So a Tupi country (located in Brazil) with legends of great riches somewhere to the north-east would end up in Europe. As would an Inuit trying to find the origin of the Vinlanders. As would an Nahua in search of Quetzalcoatl, and so would a Cueva civilisation that, after a Chinese expedition left them and further communication in that area was blocked by the Hawaiian Empire, realising that the world is round, set course east instead. Whatever reason you imagine for your alt history civilisation to cross the Atlantic, if that reason has them set course for a specific village in Ireland, then do not worry about the currents or their point of origin. If they have the technology to even cross the ocean, then they will end up wherever they want to end up.