It sure seems odd to equate Anglicanism with religion, as though all the others aren't.
If you exempt the USA and Ireland, the Anglosphere pretty much boils down to the UK, Australia, New Zealand, and parts of South Africa, and Canada. Canada is I suppose about as religious as the USA is, in much the same way--mostly Christian, just not particularly Anglican/Episcopalian. But lots of Catholics and loads of other Protestant denominations besides Anglican, with some Eastern Orthodox thrown in here and there, and a fair number of Jews and Muslims.
I don't suppose Australia or New Zealand differ radically from the USA in this respect either. Compared to the USA they might seem substantially less dominated by an ostentatious form of Christianity but I suppose about the same proportions of people actually go to services or otherwise take some form of devotion seriously.
So really, who in the developed world is more seriously 'religious' across the board than the Anglosphere, with or without including the USA?
Now implying that the only religion worth talking about is the Church of England strikes me as rather monstrously insensitive to say the least, so I have to suppose we have some weird translation issue going on here, something along the lines of having meant to say "Make the official state religion of Britain have a much larger portion of followers in the Anglosphere." That would also help explain why Ireland and the USA are exempt. Very well, to do so would probably require the government of the UK to have been more repressive on religious matters. After all, is the "Low Church," including the Congregationalist Puritans, Church of England or not? If you mean a broad spectrum of subdoctrines which are approved by the monarch and Archbishop of Canterbury, I am not sure how historical that is--whether Puritans were considered CoE or not I mean. Anyway that leaves the various dozens of other denominations, large among them the Wesleyists/Methodists, not to mention Baptists and Salvation Army and so forth all out in the cold. Not to mention Roman Catholics.
And what about the Church of Scotland, if there is such a thing? Is the Calvinist order there compatible with "Anglican" or not? (Is the British monarch deemed the head of the Scottish church as well as the English?) At the very least the UK must tolerate and support whatever religious order is dominant in Scotland as well as England, so the British Empire would have at least two denominations.
But I think that the British movement toward tolerance, at first of all Protestant denominations and eventually of Catholicism, and in the interstices non-Christian religions as well, was important to their being able to build a global empire in the first place. Maybe not utterly crucial but I think that if Britons did not adopt the notion that other people could have their own religions and it was not the duty of the British state to change their ways had a lot to do with the British being able to get their tentacles into places like India or China, and without that kind of leverage the wealth enabling them to conquer outright other territories would have been less, making for a slower and probably less comprehensive in area growth of Empire. The Anglosphere in other words would be smaller. Colonies would be less attractive places for non-British European immigrants to settle, and thus growth would be limited versus OTL. Conceivably without the policy of religious tolerance, British industry might have been slower to develop. But diversification of British piety away from the High Church of England and toward Dissenters of various kinds is the logical outcome of that, especially since the CoE was not geared to the concerns of the working class masses developing in the industrializing kingdom of OTL; there are reasons Anglicanism tends to be a middle and upper class thing whereas evangelical religions tend to flourish among the poor working classes. CoE might have had solid enough mass appeal in the traditional English countryside (though there too, I think actually Catholicism tended to hold on despite vigorous efforts to root it out as a criminal and treasonous conspiracy, so maybe even in its heyday CoE was more a gentry thing) but the Church would have had to change quite a lot to become the first choice of the industrial workers; there are reasons Wesleyians made a lot of converts there.
So basically a more monolithic CoE regime that aggressively minimized Dissenter factions and Catholicism alike would probably have so many trappings of "state religion" along the lines of Continental official faiths of various political bailiwicks that by the 20th century, instead of converting to more hard-line Christian denominations, the public would probably generally fall into religious apathy and outright atheism, more on Continental lines.