There are a number of complexities here.
First, at this time doctrine was still in flux. So it’s not as if there was an established position that suddenly changed at Chalcedon. There were a number of attempts at formulating Christology that had changed over time. Chalcedon tried to agree on a formula that allowed moderate versions of the two major approaches.
Second, the positions weren’t held by churches, in the modern sense. No one thought of separate denominations, like EO and OO. There was a tendency for different areas to hold their own theologies, but there were also different schools, and well-known individual theologians. Part of the problem is that there was politics between some of the bishops, and at times they tried to find reasons to accuse the other of heresy. There were also “dirty tricks.”
The orthodox one-nature position is a bit more complex than just “Christ is God.” Christology needs to take into account Jesus’ humanity. If you don’t do that, you’re docetic. None of the major schools intended that. E.g. Eutyches taught a form of one-nature theology that everyone rejected. Quoting from Wikipedia: “Eutychianism, which has been considered an extreme form of monophysitism,[5] holds that the human and divine natures of Christ were fused into one new single (mono) nature: His human nature was "dissolved like a drop of honey in the sea."[a] Eutychianism was condemned at the Ecumenical Council of Chalcedon in 451. Eutychianism was also condemned at the non-chalcedonian Third Council of Ephesus in 475.”
Today's churches that emphasize one nature typically use the term miaphysite theology, to distinguish it from Eutyches. They accept Cyril. As I understand it, he also taught two natures, but said that they were united into one nature. Chalcedon said that they were united into one person. It seems that the one nature and the one person were functionally just about the same. At least Cyril ended up accepting Chalcedon as equivalent. They are *not* identical except for which word they use. There are lots of complexities. But miaphysite theology does try to maintain the full humanity of Jesus.
If your instinct tells you that Christ is simply the Logos, in a single nature, you might want to verify that you’re not accepting Eutyches, which even the OO reject. One question that came up later, which illuminates it is this: Are there one or two wills in Christ? If there’s just one, then he’s not a complete human. If there are two, then it seems that there is a distinct human nature, which retains its distinctiveness despited being united to form a single person (Chalcedon), nature (Cyril) or prosopon (Nestorius).