Like Rebecca, but for different reasons, I don't like to see our Queen covering her head to visit a Turkish mosque. It isn't the head covering that bothers me, though. It's the fact that she went to Turkey in the first place.
For all my - by conservative standards - "strident" feminism, I have no objection to conservative religious dress, particularly in a religious context. I have visited many Muslim countries, and have been happy to wear a headscarf and take my shoes off when visiting a mosque. I have also taken my shoes off - I am a natural peasant and love being barefoot - in Buddhist temples. In Georgia, I was obliged to cover my head when visiting even the humblest of churches, while in Uzbekistan, a Muslim country, hardly any women did. Men, too, must often dress conservatively in a religious setting, although the rules are generally applied more strictly to women.
As a Christian, I am completely in agreement with Rebecca about St Paul. He was a man of his time. Jesus, not so limited, cares far less about what is on a woman's head than about what is in it.
Coming back to the Queen and Turkey, it would be far more worrying if she had worn a headscarf outside the mosque, as one prominent US dignitary did, if I recall, on a visit to Syria. By wearing one inside the mosque, not outside, she has emphasised a distinction between the spiritual and the temporal. But this won't do. The problem is that Islam does not recognise this distinction, and her action may be interpreted as submission.
More seriously, the visit by a respected head of state to an Islamising country may seem to legitimise that country's Islamising. That is more important than the temporary donning of a headscarf.
Our Queen is generally wise and should know better. She should not have set foot in Turkey.