It is always a threat to both politics and religion to fuse both of them together: not only do the people suffer, but they lose their freedom of religion as well. It doesn't matter which religion it may be, the loss of liberty is certain and when liberty is lost, politics loses a fundamental pillar.
Let's say, for instance, a "Christian nation". The next question would be "Catholic" or "Protestant"? History is not silent about the fact that whenever one of the traditions came to power, the other suffered. But, again, it's not just limited to "Catholic" or "Protestant". The same is the case also with, say an "Islamic nation". The next question would be "Shia" or "Sunni"? Again, say a "Buddhist nation". The next question would be "Theravada or Mahayana". And, one is aware what ramifications that has. Religion and state simply cannot ensure liberty. Not that liberty is lawless; but, when religion is enforced it loses its religious spirit.
"But, what about Jewish nationalism in the Old Testament?" one may ask. Clearly, there religion and state look indivisible. Jewish nationalism certainly was an issue when the Jews asked the question about whether it was right to render taxes to Caesar or not. Jesus' answer was that one must render what was due to the other, in this case what belonged to Caesar must be rendered to Caesar (Rom.13:7). In other words, it didn't matter what religious or ideological affiliation a state may have, religion and politics were still separate. Even in the Old Testament, there is a case when a king was struck with leprosy when he tried to mix up the two realms of authority (2Chr.16:18); also, there is a case where the monarch interfered in a religious matter when it became a political issue (1Kgs.2:27).
There will always be some ideological or religious/theological approach to politics. It is impossible for politics to be scientific, after all. Even science cannot be segregated from philosophy (much to the chagrin of the logical positivists or the 50s era); we do have a discipline called "Philosophy of Science". The logical positivists thought philosophy died under the sword of science, without considering the fact that science was still a servant to philosophy; it did what ideologies such as communism, humanism, or some religion said. Science provided the weapons, but ideologies decided how to (or whether to) use them.
Thus, it was possible for Daniel to offer a Biblical perspective of history and politics to Nebuchadnezzar while still serving as a servant in the Babylonian kingdom (Dan.2:37,39). His theology of politics didn't prevent him from involvement in politics; however, when a law made by the king conflicted with his theology, he chose to abide true to faith (Dan.6:7,10). But, in no way did he try to impose religion upon the state. He knew the Kingdom of God would come, but it was not going to be by any human hands (Dan.2:34,45). To Daniel, a proper biblical theology of politics and theology of history strongly conformed to the doctrine of the sovereignty of God in both politics and history.
Thus, we do have a "Christian theology of politics" or let's call it "Christian politics", not in the sense that it was a politics in which Christians are engaged, but in the sense that it is a Christian view of politics - and, there have been various theologies of politics (e.g. Augustinian, Thomistic, Lutheran, Anabaptist, etc). Similarly, there are also various Islamic approaches to politics.
Luther's political theology of politics and the church being two separate realms doesn't allow the concept of a "Christian Nation" anymore. Politics is secular (this-worldly).
However, what if we keep the ideas of "nation" (as people of a country - not according to race or religion, but according to citizenship) and of "state" (as a political system) separate, then can one speak in terms of "Muslim nation", "Christian nation", and "Hindu nation" -- without meaning "Muslim state", "Christian state", and "Hindu state"?
Let's say "America is a Christian nation but not a Christian state, because Christianity is not the state religion of America". Well, if "nation" has nothing to do with politics at all, i.e. one doesn't use the term "nation" in the sense of a political nation, then there is no reason why one can't speak of a "Christian nation" or even of a "Hindu nation". However, generally that is not the case. When one talks of nation, the idea of a political nation does come into the mind. And, of course there is nothing like a religious nation (if it were there, it would not respect political boundaries and so the political adjective becomes meaningless). A nation can only be political, and nationality is political, not religious. To use the term "Christian nation" or "Muslim nation" or "Hindu nation" is to categorize nation under a particular religion, as a religious nation. That is a contradiction in terms and so is illogical and unbiblical as well. On the other hand, one can safely use terms like "Indian Christians" or "American Christians", since such terms carry the idea of individual religious affiliations of citizens in a country.