I'm a Brit and have lived in the US for 10 years now. While the US has the more secular constitution, culturally and politically religion is far more influential in the US than the UK. Since the late 1970s the Republican Party began courting white evangelicals and it is now heavily associated with their beliefs. Republican Supreme Court Justices have opened the way for Republican controlled states to ban abortion and I expect we will see pregnant women die in these states as they are unable to access appropriate healthcare much as they did in Ireland until relatively recently.

The secular nature of the US constitution was designed to protect religious freedom while in the UK religion was subjugated to the interests of the state - the Church of England was setup so that the King could get a divorce after all!