Short answer: politics, identity, and the role of the state and big institutions.

Long answer:

Secularization theory has a rich body of writing. The first place you should look is Callum Brown's The Death of Christian Britain (Oxford, Second Edition 2009). The work of Steve Bruce is also relevant. Try Secularisation: In Defence of an Unfashionable Theory (Oxford, 2011).

The short summary of this theory is that secularization is a mainstream phenomenon. Church attendance has had massive declines from the 1960s onwards in most wealthy, educated, urbanised countries other than the United States. Canada is a frequently-cited example of rapid decline after extremely high church participation. The basic idea is that religion occupied a key place of prestige in society, but technological advancement, complex educational and intellectual achievement, and the elevation of other meaningful institutions caused a decline in the status and power of organized religion. The decline of religious participation does, in this construct, equal a decline in religious belief. Organized religion is an institutional construct and needs powerful organized churches to sustain it. People are less religious know because they don't find the same meaning in organized institutional churches.

The counter-argument is that secularization theorists place too great an emphasis on church attendance. They draw a straight link between church attendance and individual belief, whereas other scholars, such as Grace Davie in "Believing but not Belonging," would argue that such a metric is insufficient to understand the personal experience of religion. Church attendance doesn't measure faith, it measures social conformity. The secularization phenomenon is actually a change in societal expectation--faith and spirituality still flourish, they're simply not defined within institutional boundaries.

Everybody agrees that the big story is the rise of the so-called "nones," the people who report no religion on a census or similar form. The question is what that means for society and individuals.

In terms of the United States, firstly, it's a huge country and there are highly divergent patterns of church attendance within it. It's impossible to compare the US to the UK, given the population differences and wild disparities of ethnicity, media consumption, and the like. The UK is much more centralized.

However, within the US, over the second half of the twentieth century, there's been a steep decline of church attendance within what Americans call "mainline" Protestant denominations in the urbanized, educated, wealthier Northeast: the Episcopal Church, the Congregationalists, the Methodists, the Presbyterians, mainstream Baptists. Importantly, these congregations are all descended from British Christianity, institutionally: the Episcopal Church is the sister church to the Church of England (although, if we want to get into the weeds, they also have a very important lineage with the nonjuring Episcopal Church in Scotland), the Congregationalists are, in New England, the direct descendants of the original puritan settlers who were the Low Church wing of the Church of England and developed into so-called Independency, the Presbyterian churches descend from the Church of Scotland and its associated congregations in Ireland, while Methodism broke from the Church of England at the end of the eighteenth century and into the early nineteenth century. So in the part of America that most resembles the economy, educational level, and social cohesion of the United Kingdom, the Christian Churches most connected to British religious traditions have experienced declines similar to the one you discuss.

Why does religion seem so prominent in America? Well, and here I'm moving a bit out of my academic comfort zone and would want somebody else to write in more detail, there's been a huge shift within American religion over the second half of the twentieth century and into the twenty-first. Mainline denominations have declined overall, but in other parts of the US, more charismatic, evangelical, and otherwise more previously-marginal churches have grown in strength. In many cases, these churches represented an important force of social cohesion, social support, and personal identity for people who were not in power--the most powerful example is the Black churches, such as the African Methodist Episcopal Zion Church, but churches that supported poorer Southern white people were also meaningful to those families, and Catholics in the Northeast. There's been a huge shift where nondenominational Christianity and "looser" Churches have grown in power at the expense of the older, wealthier, more socially elite congregations mentioned above: the Mainline Churches.

In Britain, you'd have to compare the religious experience of ethnic and religious minorities to the Church of England--have Sikhs experienced a similar decline? Muslims? BAME participants in Christianity?

In short, all aspects of the secularization debate ring true here: as traditional institutional churches lose power, people become less religious: that's the decline in Northeastern and Mainline religion. But the alternative, that people are no less driven by faith and inclined towards religion and just experience it elsewhere, is also manifest in the flourishing of nondenominational Christianity and alternative religious movements.

There's also a huge point to be made about the role of the State. The Church of England is an Established Church. There's none in the US (although some individual states had them earlier in their histories). Again, it's important to think about the decline of institutional power and prestige and also the search for meaning beyond those frameworks. If there's a general loss of power in the establishment as a whole (Oxford and Cambridge, Eton, Harrow, Winchester, the House of Lords, only Received Pronunciation on the BBC), what does that do to a Church intimately connected to that power setup? Look at Robin Gill's The Empty Church Revisited (Abindgon, 2003) to look at the history of the perception of religious decline in the UK. British Churches weren't built to support their actual congregations: they existed for all, if anybody wanted to go, and were always bigger than needed. The only moment of a religious participation census in the UK, in 1850, was methodologically flawed but showed that, at the supposed height of Victorian religiosity, less than half of the country actually went to Church on a given Sunday.

Conversely, religious identity is a key driver of political identity for many Americans in a way that it hasn't been for Britain in centuries. So we're back to Churches providing a form of identity and expression to people who feel marginalized for lots of complex reasons.

In short, yes to all. Religious participation has declined in Britain, but that doesn't mean people are less religious--those who still are ardent Christians are possibly more religious than in previous generations, because they're going against a trend. The US examples are bound up in political and identity expressions that do in fact track with another aspect of the secularization debate.