You list G major and E minor as possible tonics, but I'm going to go a different direction: I hear this in A! As such, with one sharp in the key signature, it's really more of an A Dorian. Thus the progression is i v VII IV.
(Perhaps the very ambiguity of tonic was intentional, considering what takes place in the movie and the title of the piece.)
The i moving to v is clear enough, I think; it's just tonic moving to dominant. The only odd part is that the v is minor, but this is standard in the Dorian mode.
The VII is really a dominant substitute in Dorian, so that chord makes sense as well. At that point, Zimmer just moves around the circle of fourths to get back to tonic: VII moves down a fourth (=up a fifth) to IV, which then moves down a fourth (up a fifth) to i.
Note that that's different from the typical circle of fifths which goes down a fifth (=up a fourth). This may be Zimmer's popular music influence, since the circle of descending fourths is very common in that repertoire.
Another, perhaps simpler, explanation is as two related perfect fifths: i moves to v, and that ascending fifth is moved down a second to create VII moving to IV.
We can extend this further theoretically, if we want to, by saying that the upper fifth is just an extension of the original chord. In other words, the v is just an extension of i, and IV is just an extension of VII, so we can reduce this progression to just i VII, which is a very common move.
