When it comes to entering a church, halakha makes no distinction between the denomination of the church, although it may be less of an issur to enter a Unitarian, Jehovah’s Witness, or Mormon church, since these groups do not believe in a Trinity and do not believe Jesus is divine.
Anglicans also have statues and whatnot, and the Rabbanut actually sends R’ Shear Yashuv Cohen to important functions in the Vatican.
The psak is as follows. The Meiri believes that Christianity is not avodah zarah, which is like Tosafot, who say that Trinitarianism is like goyim being mishtateif HaShem and another being, which for them, is not assur, but is assur for Jews. (Tosfos sv Shema, Bechorot 2b).
Meiri says that Christians, like others “possessed of religion,” are indeed like bnei noach (Beit haBechira, Avodah Zarah 26b).
Additionally, Hullin 13b says that akum outside Israel really aren’t idolaters, but are merely following customs without understanding what they are doing.
I have read many responsa on the topic, and honestly, most poskim do not allow entering a church. However, Teshuvot haRosh klak 19, number 17, says that a Jew may take refuge in a church in a time of danger. This may serve as a limud zechut for the Rabbinate’s practice, as we would want to avoid eivah, and it is clear in this teshuva that entering a church to save Jewish life is mutar.
Historically, we do know that rabbis entered churches. Kerach shel Romi says that R’ Yisrael Moshe Hazan and others entered churches to learn niggunim that would be applied to synaggoue worship, which may be plausible, since much of nusach is pentatonic, much like Gregorian chant.
R’ Yosef Carlebach, the chief rabbi of Lubeck, often visited cathedrals for the sake of viewing the artwork and whatnot.
I believe that while it is most likely assur to enter a church, there are cases where it is simply unavoidable, as we need to avoid eivah. Also, in cases where Jews have gentile family members and colleagues, it may be detrimental to avoid attending functions like church weddings, funerals, christenings, etc.
The British Rabbinical practice of entering Westminster Abbey, like that of Rabbi Haskel Lookstein at Obama’s inauguration, is undertaken for the sake of good relations with our neighbors, which is a religious imperative in and of itself.