For Asian, you could have Japan drop its isolationism earlier on and adopt some technology and other things imported from Europe via the Dutch and so on. Have a strong emperor who cracks down on the power of the nobles (or hell, even a really powerful shogun) and decides to establish colonies, European-style, while America isn't fully claimed yet. They could grab a few islands in the Pacific and maybe even parts of America - California, or what is now the Oregon region. If Japan successfully modernizes in that age, maybe China can follow suit.
Africa is harder, as others have mentioned, especially with a PoD after 1492. If you allowed a PoD within the 15th century, maybe we can get someone in Morocco to follow Spain and Portugal to the Americas. That is, if Morocco counts as "African". Other than that, no one in Africa really has the resources to rival European powers before they claim pretty much everything in America.