Milt:
What a question!
No-one has any approved schools for astronaut training. Not sure there is a set of standards that could be used for getting an astronaut licence (notwithstanding the new FAA licence given to the Scaled Composites pilots).
In addition to the 'military' schools, National Test Pilot School in Mojave is 'approved' for experimental test pilot training.
If a test pilot course is run, it pretty much always includes a flight test engineer course.
As for other countries - many run test pilot courses as they have a need. Brazil appears to have more or less annual courses, as does India. As for approval, the Society of Experimental Test Pilots (SETP) has a way to 'approve' courses based on content, aircraft, etc.
Hope that helps.