You honestly try to make that argument without serious hyperbole?
The UK definitely doesn't have separation of church and state - it's got an official state religion. Although it is in practice pretty secular. A quick look around various countries (see Gethsemani's post) would quickly reveal that the influence of religion upon the state is far more substantial in many Christian countries than you suppose.
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What you're really arguing is about a degree of development in cultural and societal attitudes. "Christianity" hasn't so much gone through the enlightenment as Europe has - and by extension some of its ex-colonies also. Plenty of African countries are Christian, and show very little presence of Enlightenment values. Lots of Christian, ex-Soviet states have been backsliding on democracy and some are even unrecognisably democratic.
The Enlightenment was basically the entire 18th century - 100 years of ideas and social development. And then after that 1800 was hardly the living, breathing embodiment of secular, humanistic ideals, and 1900 left a lot to be desired. France deposed its monarch and became a republic in ~1790, and took another 80 years before democracy finally stuck. It took literally centuries for people to gradually adapt. Think on it this way: people are brought up with ideals as children, and these stick with them to a large degree for the next 50-60 years of life. That's the time frame we need to think about. Widespread societal change in a country is usually the work of generations. Why on earth would anyone think the Islamic world, or Africa, or other places were magically going to turn into secular, liberal democracies in a few decades?
If we compare for instance Turkey and the ex-Soviet states, support for secularism is very high (~80%+), especially compared to the Arab states. Of course it is. They had 100 years of society bringing them up with under secularism. Likewise Western Muslims, despite mostly originating from states without a culture of secularism, show vastly higher support for secularism than the countries they or their ancestors came from. Of course they do, they've been brought up with secularism. As the years roll on, they'll become even more secular. The people who say "Islam is incompatible with Enlightenment values" are simply people who haven't waited long enough to see it finish, and refuse to accept the evidence of the process occurring right in front of their eyes. Like there's a glass gradually filling up, they see see it around the half way point and say "nope, still empty".