I am rewatching The Tudors, and there is an episode where King Henry is in a jousting accident. Plans are made to declare Princess Elizabeth as Queen in her minority, with Anne Boleyn as Queen Regent. At the time, Anne is pregnant. Had Elizabeth been declared queen and Anne given birth to a boy, would the coronation be reversed because of primogeniture?
And yes, I know The Tudors is dramatized, so this question is purely hypothetical. But it also could have happened to Mary Queen of Scots, as she was extremely young when she ascended, and her mother was still of childbearing age. I'm sure there are other examples.
For a twist, could this have happened in places other than Western Europe as well?
You are probably thinking of this as a legal question; in reality it is a military one. As in, "it depends on who wins the ensuing civil war". There would always be someone powerful ready to back the child who was already born (and become Regent, which could be a stepping-stone to higher titles) and there would always be someone ready to back the king's son (and become Regent, which could be a stepping-stone to higher titles). Both sides would be able to find some kind of legal backing, but that would be window dressing; the issue would be settled entirely by their respective armies.
In 1316, Louis X of France died, leaving behind a daughter, Joan, and a pregnant wife. His brother Philip ruled as regent for five months until Louis' wife gave birth -- to a boy, as it turned out, who inherited the throne on birth as John I ('the Posthumous'). As he only lived for five days, though, the question of how well his claim to the throne would have survived was never tested.
So yes, legally there was a case, at least, for a posthumous child to inherit ahead of an heir presumptive.