So I was just now considering how colonization would go so here's my ramblings on that front
One of the big things that caused colonization of the Americas IOTL is that Europe didn't have nearly as many luxury resources as Asian lands like China and India did (Christopher Columbus went west specifically because the Spanish wanted spice that the Ottomans closed off to them). Without such a motivator, colonization is gonna be slower.
Also, American geography needs to be taken into account. Most of the large mountain ranges in the Americas tend to be on the Westward side (the Rockies and Andes), and the few on the Eastward side (Appalachia) tend to be smaller and shorter. And that's not even getting into the sheer size of the Pacific Ocean, so rest stops like the Alaskan islands, Hawaii, and Polynesia are gonna be very important.
Asian colonizers would have an EXTREMELY rough start, not only having to pass over the Pacific, but also having to cross extremely large mountain ranges or even further sea trips to get to the arguably much more settleable East. The Northwest Passage and southern tip of SA are gonna be crucial, with whoever owning them essentially holding the keys to the Americas, and there could even be an Alt-Panama Canal earlier than OTL out of sheer desperation by the vying powers. If there's an Alt-American revolution, it's gonna happen way later than IOTL, since it'd take way longer to get settled, but a USAnalogue is a rambling for later.
Not to mention, prosthelytizing their faith in the name of God. I have no clue if there'll be a religion that's as pro-expansion as OTL Christianity. Then again, if Christianity is essentially Buddhism, then it's likely that Buddhism is the Christian analogue with Buddha and Jesus swapping roles (or more crazily, there'll be an entirely different AltJesus). Nationalistic Buddhism IS a real thing, just very hard to find (see Burma's relationship with Muslims), so it could work. This would also mean that East Asia completely embraces Buddhism while Christianity is semi accepted but not the norm in Rome. They'd probably keep the old pantheon around.
I could keep going on implications for a long time, but I'll stop here. While this is a swap scenario, it's definitely not a 1:1.