How can I convert a non-ICAO (Taiwanese) pilot's license to FAA or EASA? So I'd like to know from somebody what's the process of converting a non-ICAO license. In particular, I'm trying to understand how a Taiwanese PPL can be converted to FAA PPL (I personally know people that have done it).

Taiwan is not a member of ICAO due to it not being recognised as a country due to China. So does that mean that a Taiwanese license is considered as a Chinese one and maybe that's why the FAA does the conversion?

I'd also be interested to hear about anyone who has tried to convert to EASA. I've never heard of anyone doing that.
I can't tell you anything about the EASA regulations, but there seems to be no way to convert a non-ICAO license directly to an FAA one. 14 CFR 61.75 says (my emphasis):


  (a) General. A person who holds a foreign pilot license at the private pilot level or higher that was issued by a contracting
  State to the Convention on International Civil Aviation may apply
  for and be issued a U.S. private pilot certificate with the
  appropriate ratings if the foreign pilot license meets the
  requirements of this section.


As you said, Taiwan is not a signatory to that convention, so the FAA can't issue a license under 61.75. (61.153 allows converting a foreign ATP license to an FAA one, but it also requires that the license come from a contracting state.) I couldn't find any special regulations or exemptions for Taiwan on the FAA's site.

You said that you know people who have "converted" a Taiwanese PPL to an FAA PPL, but do you know exactly what they did? My guess is that they applied for a regular FAA license, and used their flying time in Taiwan to count towards it.