Does religion have any relevance in politics and vice versa?

No. The US was founded upon the concept that began with ancient Greek city-states, that religion should play no part in the the political (governmental) decisions made in the best interests of the citizenry, as enshrined in the Constitution (that the Right like to cherry-pick, as they do biblical text, for their own purposes) to incorporate de Montesquieu's secularist proposition of the separation of powers. The rise of fascism within America can be seen to walking very firmly hand-in-hand with the fundamenatlist, exclusive and uncompromising theocratic impulse of the Christianity the Founding Fathers worked to exclude from civic decision-making.

The Right hijacked Christianity some time ago, as a means to advance the notion of "identity" politics (putting religious decrees, as in the prohibition of safe abortion, ahead of political concerns) that persuades many of the electorate to choose to vote against their own economic interest, instead voting to oppress, for example, the LGBT community in a classic case of divide and conquer.

Like all aspects of our lives - we should base our important descisions on a rational basis. Critical thought should be brought into play and policy should be set with due regard to the evidence not doctrine, baying mobs or scripture.

No, none, zero, and should not !!! Unfortunately, the current political climate in the world is so heavily influenced by bogus beliefs, it’s become totally uncomfortable.
Such hypocrisy, un-effing believeable !

Calling the UK a Christian country is technically true, though population wise the majority of the populous are non practicing Christians but the church of England does of course have a vastly disproportionate influence on UK politics.
You are actually mistaken the Arch Bishops and bishops (26 in all) of the church of England have a seat each in the upper house of parliament (House of Lords) and a vote on the passing or reconsidering of legislation. Also the Queen has a royal veto, it is rarely used but does exist.
The UK is in this respect one of the only four western theocracies.
UK/commonwealth, Israel, Utah, the Vatican.