Religious background - whether or not it has been renounced - clearly plays an important role in determining the shape and tenor of a person's political and social views. Even people who have not had a religious upbringing are often influenced by religious elements of the broader culture.
I am interested not so much in survey data etc. about links between particular religious traditions and particular political ideologies (interesting though this can be) as in the logic behind the links. For example, it seems to me that the left owes a huge debt to Judaic sources insofar as its basic project is an attempt to make real a religious vision of a new earth, a promised land of harmony and prosperity.
Protestant churches and sects are generally closer to Christianity's Judaic roots than churches in the Catholic tradition (e.g. Roman Catholic, Eastern Orthodox, Episcopalian and Anglican). These latter traditions are not so much Christian as Christian-Platonic, following Plato in seeing the soul as essentially spiritual rather than essentially embodied and earthy. Judaic notions of the resurrection of the body and a (mass) last judgement sit uneasily with the belief of most traditional Catholics and Episcopalians in a soul that leaves the body at death and makes its individual way to heaven. (The Pythagoreans and Platonists believed something like this.)
But, ultimately, all traditional Christians and religious Jews believe in a spiritual - or at least in a supernaturally transfigured - realm, and such a belief is very compatible with political conservatism (and with political quietism which could be seen as a non-activist form of conservatism). Clearly, for religious people the main game is to get things right for the long haul - for the spiritual or transfigured realm - and secular institutions, political or otherwise, are of secondary importance.
Those who have lost their faith in a supernatural solution may or may not retain the moral priorities and ideals of the renounced religion. They may, like Nietzsche, find inspiration in the aristocratic values of classical Greek and Roman culture, so different, as well-known passages from the New Testament and related works make clear, from Jewish and early Christian social and moral teaching. Though non-religious conservatives will have various views on these matters and may retain many elements of Judaic or Christian ethics, they will not, as a rule, attempt (like the non-religious, left-leaning liberal) to implement a secularized version of Judaic or Christian moral and spiritual ideals.
In this the conservative - non-religious though he/she may be - shows more realism than his/her 'progressive' equivalent, and, in fact, a more profound understanding of the holistic nature of religious thinking.
Of course, the dramatic rise of Islam and, in particular, of militant Islamic fundamentalism, has changed the whole dynamic of the interplay between religion and politics in the West. But that is another story. For now I merely observe that these developments have highlighted the close links between Western religious traditions and our more general notions of freedom. The exploitation and abuse by Islamic extremists of Western conventions of religious freedom not only put those conventions at risk, but, with them, other freedoms which we have taken for granted but which are in fact the delicate fruit of a long historical process.