From all accounts, religion in the UK (barring the two Irelands, perhaps) just isn't much of a hot button issue, either in government nor in personal affairs.
I never like seeing governments offering religions favors of any kind out of their status as religions alone. That amounts to giving the State the power to decide what should count as a proper religion, and in practice opens the way to favor and disfavor specific ones among those recognized. Government initiatives should utterly ignore any claims of being a "religion" from any group.
The House of Lords is an odd duck, from what I understand. It holds real but usually not decisive power, and it is very much beyond the boundaries of representative democracy. I tend to think of it as a retirement home of sorts for politicians that some government decided to reward with lifetime status and salaries. Very much an anachronism, but also a living (if expensive) embodiment of traditions with some value as a source of advice. It has provided an outlet for voices that apparently just won't come from those who are aiming to be elected or re-elected, and in that role helped in the stability and democracy of the UK.
From the perspective of the people themselves, apparently the British largely consider religions, creeds and beliefs a personal matter that is rarely questioned outside of close proximity relationships or, oddly enough and going to the other extreme, in matters of very wide public interest. Apparently it is proper and perhaps necessary for members of the Royal Family to be members of the Church of England; literally illegal for certain members of it to be Catholics; and adherence to other creeds is perhaps seen as mainly a personal eccentricity of little consequence. The bloody conflicts between Protestants of the CoE and Catholics are well remembered (particularly in Ireland), but not much of an actual current political issue. Other religions do not seem to be politically relevant, except perhaps as a component of the larger issue of immigration.
The Church of Scotland is even less controversial; it has official acknowledgement and clearly it holds special significance for Scotland, but has far less privilege. By my reading of the current versions of relevant Wikipedia articles, about one in each three Scots considers himself a member, but only one in twenty actually attends services regularly.
It seems to me that both Churches are mainly preservers of heritage and tradition at this point, a situation that has both advantages and downsides, the later being mostly of a financial nature.
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