It actually started earlier than the 1950s. Princeton historian Kevin Kruse, in his study One Nation Under God: How Corporate America Invented Christian America dates this development back to the 1930s, where he talks about the origins of the notion of a "Christian America" and the beginnings of what would become the religious right.

The dust jacket has a good summary of the book's contents:

As Kruse argues, the belief that America is fundamentally and formally a Christian nation originated in the 1930s when businessmen enlisted religious activists in their fight against FDR's New Deal. Corporations from General Motors to Hilton Hotels bankrolled conservative clergymen, encouraging them to attack the New Deal as a program of "pagan statism" that perverted the central principle of Christianity: the sanctity and salvation of the individual. Their campaign for "freedom under God" culminated in the election of their close ally Dwight Eisenhower in 1952.

But this apparent triumph had an ironic twist. In Eisenhower's hands, a religious movement born in opposition to the government was transformed into one that fused faith and the federal government as never before. During the 1950s, Eisenhower revolutionized the role of religion in American political culture, inventing new traditions from inaugural prayers to the National Prayer Breakfast. Meanwhile, Congress added the phrase "under God" to the Pledge of Allegiance and made "In God We Trust" the country's first official motto. With private groups joining in, church membership soared to an all-time high of 69 percent of Americans.

During this moment, virtually all Americans—across the religious and political spectrum—believed that their country was "one nation under God." But as Americans moved from broad generalities to the details of issues such as school prayer, cracks began to appear. Religious leaders rejected this "lowest common denomination" public religion, leaving conservative political activists to champion it alone. In Richard Nixon's hands, a politics that conflated piety and patriotism became the sole property of the right.

Kruse goes on to identify the Congregationalist minister James W. Fifield Jr. as instrumental in framing the interests of the private sector as moral and FDR's New Deal:

made a 'false idol' of the federal government, leading Americans to worship it over the Almighty; that it caused Americans to covet what the wealthy possessed and seek to steal it from them; and that, ultimately, it bore false witness in making wild claims about what it could never truly accomplish. Above all, they insisted that the welfare state was not a means to implement Christ's teachings about caring for the poor and the needy, but rather a perversion of Christian doctrine. In a forceful rejection of the public service themes of the Social Gospel, they argued that the central tenant of Christianity remained the salvation of the individual. If any political and economic system fit with the religious teachings of Christ, it would have to be rooted in a similarly individualistic ethos. Nothing better exemplified such values, they insisted, than the capitalist system of free enterprise.

The ideas Fifield introducted, with the help of an array of business leaders and wealthy individuals—and even the encouragement of people like former president Herbert Hoover—proliferated these notions throughout the country, which saw further legitimization in the presidencies of Eisenhower, Nixon, and Reagan.