So what would a membership of NUS mean? It almost certainly would mean more possibilities to influence countrywide decisions concerning students, whilst local concerns might not be as much of a concern or worth a campaign by the NUS.

The Edinburgh University is regarded by many as too conservative and old-fashioned, as well as not progressive nor tolerant enough.

Would or could this even change with facilitated by membership of the EUSA in the NUS?

NUS has quite a radical left history, but in the last years, with the "new middle" in the Labour party NUS also has shifted closer to the centre-right.

So, would or could the political awareness of students in Edinburgh even change by membership of the EUSA in the NUS and move to the left?

This is not sure, but the “yes” campaign stresses the impact of a membership in the NUS with a potential increase in local campaigns for ethical, environmental and global issues.
As an example the “Yes” campaign presents the campaigns against Nestle and Coca Cola, GlaxoSmithKline and Coors, which were also facilitated by the NUS.
NUS runs a strong anti-racist, anti-fascist campaign. They run campaigns against the BNP, especially when they try to organise on campuses across the UK.
NUS has campaigned for many years for equal rights on issues of sexuality, gender, race, religion and disability.

Article in the Edinburgh Student newspaper:
 http://www.studentnewspaper.org/view_article.php?article_id=200410251611