Show me any sort of political movement before the contemporary period that hasn't had some sort of religious aspect. All those movements that you state may have had religious influences, and certain principles derived from religious morality, but they weren't religious movements. In fact the only one of those that you listed which came close to being overtly religious was prohibition in it's early stages.
My point being, you are not going to get politics aligned in the way supposed by the original poster without serious, serious disruption. The left may have drawn it's principles from religious inspiration from time to time, but that is a damn sight different to having a left dominated by religion and a right opposing that and championing secularism. That is in contradition of at least two hundred years of history.
Just dealing with America, one of the most religious Western world societies, the abolition of state churches actually followed a period of greater religious fervor across the populace, as well as more direct church involvement in politics. Perhaps you are confusing hierarchy for religiosity, because that is one expression of religion we have largely shied away from.
Prohibition was not the only such movement. In fact, it was a movement where churches were less involved. It mainly got hijacked by suffragettes and later on, businessmen. It started however, in one of the Great Awakenings.
Abolitionism was very much a movement of the churches and by clergymen. In fact, the often zealous tone and nature of abolitionist churches led to a Southern condemnation of that very type- because abolitionism was a primarily church-led movement of fanatics, it was thus a flawed and irrational idea. Besides which, ever hear of Beecher's Bibles? I imagine the folks who distributed those would disagree that it was not a religious movement.
An argument could be made that much of the work that lasted during Reconstruction was due to the zeal of the above movement. (Churches were made and became the centers of black communities, school teachers were more often than not, from abolitionist backgrounds, etc....)
Populism, once again, did involve the churches all across the South. Many of their early orators and rabble rousers were churchmen. William Jennings Bryan was a frontier preacher at a time, and used ideals of Christian equality and justice while on the campaign trail. The Populists championed good church values, a day of rest on Sundays and enjoyed a wide range of support from the religious and the church institutions, especially in the South.
Progressivism's religious arm grew out of a condemnation of the "gospel of wealth" and the early settlement houses used and cooperated with the churches to spread their message and do their work. The Progressive Era was highlighted by a growth in inner-city churches and an expansion of social work by the churches. American Christianity grew condemnatory of capitalism.
The New Deal (well, the Great Depression) saw yet another growth in the style of the Progressive Era. There is a reason that Jimmy Carter's election was the last gasp of the religious Southern Democrats- they had been indebted to the Democratic Party since the New Deal. There are of course, counter examples here: Charles Coughlin and the like, but many churches fell behind the New Deal.
And if you are to claim that Civil Rights did not involve churches well......
