The impact of not having Europe colonise parts of the world would have a such large impact on culture that it's literally impossible to write it all in a thread. Overall, world culture would be more diverse. Christianity wouldn't be the biggest religion, but multiple traditional religions would flourish in Africa and Asia. Africa especially would be significantly richer overall. The diseases introduced and the genocide committed by Europe (and by extension, European colonial successors) in the Americas would not have occurred, so Native American culture and language would continue. With no imperial force demanding absolute authority, cultures would not have to Westernise, so even countries that weren't directly ruled by Europe in OTL would be significantly different. Even in Siberia, we would see much greater diversity in culture and language. Capitalism would not grow to be as dominant outside of Europe as it did in OTL, and other ideologies that our world never had the chance to develop would certainly develop in this world. Slavery on an industrial scale such as in the Americas would not have occurred. Most countries in OTL would not exist ITTL.
I think you're being a little too idealistic in your assessment of how a lack of colonialism would change the world. For one thing, the challenge could very well be accomplished if, say, China undergoes an Industrial Revolution under the Song and forestalls European colonization; but clearly this would most likely end up with merely a replacement of European imperialism with Chinese imperialism. While that might be different in the details the experience of Korea, Vietnam, and Japan, not to mention Mongolia, Manchuria, and Tibet suggests that you would still see heavy adoption of Chinese models, at the very least, including culturally and religiously. There's no obvious reason to suppose that Sinicization is better than Westernization, unless you're a massive Sinophile and Europhobe, so this scenario would both fulfill the OP's request and completely
fail to fulfill your predictions.
For another, I think you're underestimating the effect that
peaceful,
non-colonial contact and trade can have on cultures, languages, religions, and society in general. To continue with the China analogy, Japan was obviously never colonized by the Chinese, unlike Korea and Vietnam, but still adopted many aspects of Chinese culture due to the intensive contacts that took place between Japanese and Chinese society. Although Japanese culture remained distinct from Chinese culture, it was still clearly Sinicized in many significant ways. Similarly, Hinduism, Buddhism, and Islam spread and displaced indigenous religions in Southeast Asia without them being colonized, due again to intensive contacts with Indian and Arabic peoples who brought those religions and associated cultural aspects into the region, where they were then adopted by local peoples. Assuming that "no-colonialism" doesn't mean "no contact" or "no trade" or generally speaking no intercourse between peoples that were historically colonized and Europe (or Eurasia more generally), I think you'll see significant cultural adjustments and borrowings from European or Eurasian models that you could reasonably describe as "Westernization," and likely a significant spread of Christianity in many regions of the world.
Finally, I think you're underestimating the effect that
internal imperialism would have on the cultures and societies of the Americas, Africa, and other places that were colonized by Europeans. I already cited the effects of Chinese influence in East Asia, but one could similarly look to the Inca in South America or the Caliphates of early to medieval Islam to see forces that attempted or succeeded in inducing substantial cultural homogenization over large swathes of land. While a lack of European colonization probably does mean that there are fewer
European-derived states and cultures in European-colonized areas, and it surely alters how those cultures evolve, I think it's naive in the extreme to suppose that there won't be other native empires that either consciously attempt to and unconsciously cause the spread and adoption of a relatively uniform culture, religion, and language over large regions. I am not sure that you will necessarily see (much) more diversity so much as
different diversity; fewer European-derived cultures, as I mentioned above, and more native cultures, but still mostly imperial cultures from post-contact states.
Also, diseases were probably inevitable in the Americas and Oceania, unless we're supposing a "no-colonialism" world means a world where sailing techniques and technologies outside of the Polynesians never grew sophisticated enough to cross the Atlantic or Pacific Oceans, which seems rather different than what the OP intended. To take an example from where I live, Hawaii was demonstrably struck by multiple plagues after making contact with Europeans, despite maintaining its independence for over a century afterwards and being questionably colonized during much of the time that the plagues were most serious.
