Christmas is a messy blend of different aspects from a vast array of origins ranging from religions to cultural traditions, to corporate inventions, to smaller scale local innovations from just a single family to a whole town or country. What Christmas means is unique to each person and can vary hugely just between two families in the same town, let alone people from different countries.

You could argue that yes, those who are not religious should not partake in the aspects of Christmas that come from Christianity (the name, the carols, nativity plays, etc.) I would suggest that there is nothing more fitting for the UK, than how we currently celebrate Christmas. Taking part in the religious side, but not for the religious reasons is very fitting. There is so much around modern Christmas nowadays which has very little to do with religion. 

For me personally, I am religious, and I love both the religious and non-religious sides of Christmas. I think that the idea that religious festivals should become exclusive is a very non-Christian idea anyway. Therefore I believe that anyone should be allowed to celebrate Christmas in their own way, and if they should want to, even the religious aspects. Also, it is the Christian belief that everyone should be allowed access to God.