It's possible to separate churches from government, but it's impossible to separate religion from government, because governments and national identities themselves are based on a large number of concepts that are deeply religious in nature. Nationalism, for instance, is a religion, whether or not it is associated with either a "god" or a church. So is culturalism.

This was abundantly evident in the case of various officially atheistic regimes in recent history such as, for instance, Mao's China or Stalin's Russian or Pol Pot's Cambodia. While they railed against organized religion, they were themselves deeply committed to a political religion of their own invention. It is also abundantly evident in all other nationalistic regimes...and that means ALL regimes. They are all deeply religious, because they deal in popular myths of their exclusivity, their sacred symbols (like the flag, the American eagle, etc), their assumed superiority over outsiders, their rituals, their traditional forms, and their rigid power hierarchies.