So in recent months an idea kicking around in my head has been a group of East Asian Buddhists over the course of several centuries making their way to Europe where they set up a monastic society in the Alps, preserving in many aspects their culture. While that idea specifically is VERY unrealistic, one thing that I have always been interested in is whether or not Buddhism in any form can make its way to Europe.
I don't mean becoming the dominant faith. Far from it. Instead, have a sizable portion (Say 5-10%) or even an entire country adopting the faith formally or informally. What are the odds of this happening? Where is this most likely to happen? When is it most likely to happen?
When is it most likely to happen? Possibly in the future, where Buddhism could take off to that degree. Or maybe instead in an alt-20th century. I don't think it would be much more than 5-10%. Either way, I think it would be from mass immigration of a primarily Buddhist population alongside Westerners taking up Buddhism.
But aside from that, maybe it could be transmitted by the Mongol Empire during the invasion of Europe? A wholly Buddhist state is unlikely to emerge, or last for very long (they'll be forced to convert before long), but the corresponding Buddhists and their converts (probably speaking Tatar or Cuman) might make up a decent fraction of the people and survive as a distinct ethno-religious group. Look at the Lipka Tatars for the Muslim example of this.
Hungary, Romania, or Poland is the best place for this. I think Transylvania might be the best place, where a Buddhist Turko-Mongolic ethnic group could migrate into the hills and effectively co-exist with the Orthodox and Catholic (and Jewish) populations. A worse depopulation of Transylvania during the Mongol Invasions would be the way you'd get this situation to emerge. Thus, I think that Buddhists could comprise maybe 10-15% of Transylvania's population, if not more. They'd be an important minority group in whichever state controls Transylvania.