Thanks for the help with my other posts on this timeline.
So far we've got a timeline where there the Mongol conquests only last for 10 years after which they have no influence on the world outside Central Asia again. The invasions of Europe are roughly similar to the OTL invasions between 1236 and 1241. At that point things turn out differently in Europe. The Mongols are expelled by an European offensive. There's no Black Death in this timeline.
A globalized trade network around some time in the 1300's and potatoes have spread around the world, resulting in much larger populations. Technology is spread around the world by the trade network so expect nearly all potential colonization candidates to have guns and roughly comparable tech. North and South America have being exposed to diseases from Asia prior to contact with Europeans and have had time to recover their populations. The world's nations are still as divided as ever so playing them off against each other like in OTL South America is an option.
Now that everybody has guns and the populations of the Americas are immune to European diseases, how does Europe as a whole develop afterwards? In this situation does Europe even begin to colonize and start building their empires? There wouldn't need to scout a route through to China, the folks in the other regions have already created multiple trade routes that can do that. Do they even have the strength or motivation to conquer any significant amount of land? I would imagine it could be similar to a snowball effect. Start with piracy on the trade routes, then go after weak targets first, the more land is conquered, the faster these empires could expand and the more powerful opponents they could engage.
If they find that individually they cannot build a empire of any significant size, would we start to see the nations of Europe forming an alliance in order to help take land? They may initially form alliances to fight each other and use these alliances to expand. Eventually when they come across the stronger and richer world powers we could see the separate alliances merge into a single alliance containing most of the European nations. This would be an 8 nation alliance style pact, where the involved nations actually are actually enemies working together temporarily to increase their own power and take out threats external to Europe.
Perhaps even through the nations of Europe a have potatoes, due to poor planning parts of their infrastructure begin overburdened and further destabilize the nation. Perhaps expansion would be in part a way to alleviate these problems.