What did Elizabethans think about God intervening in human affairs and how does this relate to the Oracle of Delphos in The Winter's Tale?
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.