They are in NATO. They aren't a major player in an world events, but their alignment means they are definitely on one side and not the other. Not much more to say for their part beyond that. Maybe someone else could tell you more, but as far as I can tell, that's all there is to know.
If you want to learn more about Cyprus, I discussed it in-depth with the Myth of the 20th Century: https://myth20c.wordpress.com/2021/03/04/cyprus-a-nation-divided/