Game time 20 minutes or actual time? I ask because I was playing over an hour.
Maybe the loop only starts once you leave the home planet? I have no idea. You need to die for a save though I think.
-- I'm not sure if this spoilery, so read with caution if you want to go in absolutely blind.
But it seems the mechanic is a bit mysterious, and I'm just talking mechanics.
The first loop ends approx. 20min after the statue opens their eyes, so if you took your sweet time in the village, then it's possible you played for an hour without the universe exploding.
Once the re-awakening happens for the first time, your progress will be saved, and as others have said, is going to be saved automatically going forward, at least the progress you're making in exploring, it won't save your location or anything.
The game for a long time doesn't really explain what's happening, so there's no countdown or anything that would indicate the 20min time frame. You'll eventually see the rhythm of the cycle which can help you understand how far along you are by how the planets have moved, which stuff is happening and so on.
It's a puzzling game for sure, but it's part of the fun after all. Thankfully, your natural curiousity (and the rumor log) will help you figure it out for yourself!