While Shakespeare wrote in the Elizabethan era, Romeo and Juliet was set in and referred to history that took place in the early 14th century, the Middle Ages. During the Middle Ages, the Roman Catholic Church dominated all of Europe and acted as a government. The Church imposed laws and taxes and even owned its own land ("Middle Ages Religion"). The religious presence in Romeo and Juliet reflects this domination of the Church. Not only are there scenes portraying some of the characters' commitment to the Church, the play especially shows the lawless disregard for the Church's ordinances, which is one factor that leads to the tragic deaths in the play.
One way in which the characters of the play disregard the Church's rule is through the ongoing feud between the Capulets and Montagues. Specifically, this feud instigated street brawls and duels, like the one we see initiated by Tybalt. While dueling was a very common and proper form of revenge, the Catholic Church opposed dueling because they saw it as a means of testing God (Catholic Encyclopedia: "Duel"). Many popes from early on ruled against dueling, even the popes of the 13th and 14th centuries ("Duel"). Hence the fighting and dueling we see taking place in the play also broke the ordinance's of the Church. Prince Escalus's own newly passed decree reflects the viewpoint of the Church, as we see in his lines, "If ever you disturb our streets again, / Your lives shall pay the forfeit of the peace" (I.i.92-93). Hence, reasons why the Church is an important element in Romeo and Juliet is, first, because the play accurately depicts the society and culture of the time, and, second, because Shakespeare uses the Church to make a moral statement about violence, feuding, and dueling.
It's hard to say exactly how religion in the play would have been portrayed had it been set in Wales today if one is not familiar with that location's culture. However, one thing which can be known is that the dominant religion in that area is now the Anglican Church, which broke away from the Catholic Church. There are not many differences between the Catholic and Anglican churches in terms of things like creed or liturgy as both started from the Eastern Orthodox church ("What is the Difference"). The greatest differences are in terms of who has central power. Therefore, if the play was written to still refer to religion and use religion as a moral standpoint, then it would still be very much the same. However, a legitimate question is whether or not religion or morals from a religious standpoint would still be important to the citizens of Wales today. The 2011 Census indicated 32% of Welsh citizens do not associate themselves with any religion, while only 18.5% do ("2011 Census"). Therefore, if the play were written to address the general Welsh population, it's very unlikely that it would contain any religious references at all.