How to prosecute plagiarism when the discoverer of the plagiarism is an outside third party? Recently a group of students have investigated the the alleged plagiarism in the thesis of one of the most famous alumnus's of Glasgow Caledonian University. In one specific case, a crowd-sourced project highlights very convincing evidence of plagiarism throughout the process. Is there any global authority to investigate this type of issue and put it through required legal steps?
There is no single global authority that deals with academic plagiarism.

While the current legal system stands, there's no quick and reliable way to prosecute anyone because everyone has a right to defend in court, and court processes are long, tedious and don't always result in a way that you expect. Your question seems to assume that plagiarism should lead to immediate dismissal of the president; this will not happen. Actually, this should not happen without a thorough investigation.

This being said, I sympathise with your cause, but you have little legal ground here. The university has the right to revoke his PhD degree, should the university decide so. If the university does not, it's the reputation of the university that is at stake. You have the right to distribute this information and show that the university officials are caring about politics more than about science.

Also, it is highly unlikely that a plagiarised PhD thesis would get a president of any country to quit. I have been closely following the work of Dissernet, a similar group in Russia that examines PhD theses for plagiarism. While they have uncovered plagiarism in theses of several high-ranked officials, including the Minister for Culture, no one in Russia bothered to admit it, much less quit, zu Guttenberg style.