The world of the play is not Christian so the Elizabethan audiences would have understood that.  Both kingdoms, Sicily and Bohemia, are worlds where an oracle is consulted and statues come to life.

When Leontes defies the oracle, he brings down the wrath of the god Apollo.

As for Divine order, the Elizabethans believed in the Great Chain of Being where you would find Divine order.  As there is a hierarchy in heaven, so too on earth.  A man would be born to rule because God wished him to be king.  From the king, the order fanned out below.

If the order were interfered with, the effects would be felt throughout the society.  The final part of Apollo's message is important, "...and the King shall live without an heir if that which is lost be not found."

A sense of order was important in the turbulent world of Elizabethan England.  Civil wars and power struggles were recent history.

Shakespeare knew how to transport his audiences to exotic places where life was different yet the same since human beings are human beings no matter where or when.

How should Elizabethan theology affect our interpretation of the consultation with the Oracle of Delphos in The Winter's Tale?

As many revenge dramas, Shakespeare`s A Winter`s Tale is set in a fictitional universe, oddly blending geographically distant Italy with pagan Graeco-Roman culture. For a king to consult the oracle of Delphi before taking a major action would have been normal procedure in 5th or 4th century Athens; the oracle itself and consultation with pagan oracles would not have been done in Renaissance Europe (though people might pray to saints or make pilgrimages to holy Christian sites).

In part, the consultation with the oracle shows the influence of Seneca, a Roman writer, on the genre of revenge tragedy; such consultations were common in both Seneca`s plays and the Euripidean tragedies that influenced them.

Christian tradition does include inescapable prophecies, by primarily Jewish prophets, e.g. the prophecy of the Magi concerning the birth of the King of the Jews leads to Herod the Great`s Massacre of the Innocents in the Gospel of Matthew. Prophecies by Jeremiah etc. are an important part of the Judaeo-Christian tradition, but the Delphic oracle is a literary device for Shakespeare (like the mention of Persephone in your other question) invoking pagan literary generic conventions.