There are a number separate issues here
The role that Christianity has in the UK because it is the "state" religion, which is the whole ongoing (for centuries) discussion about the separation, or not, or church and state. As the UK consitution stands, officially it is a Christian country and they aren't separated officialy, like they are in countries like France and the US. The head of the country (the Queen) is also head of the church, there are seats reserved in the House of Lords for Bishops, pretty much all the Public Holidays are Church related.
Then, there's the identification of "British" culture as being Christian, to varying degrees. That's a seaprate issue. You could completely separate the church from the state and ban any publicly organised/recognised religious festivals, and cancel things like Christmas which is (despite its pagan roots) seen as a Christian holiday. I bet if you tried that there would be rioting in the streets because, in the UK, Christmas is an integral part of British culture; as many people have said, lots of people who don't identify themselves as "Christian" still celebrate Christmas (e.g. my sikh boss, my atheist SIL).
Then, the other issue is people who feel that because they are Christian, they should get special treatment like the crucifix woman - there is no religious requirement in Christianity to wear a cross. It's a "nice to have". A head covering and dressing modestly is, however, required in Islam. They're separate things but linked to the issues above.