You don't know that. I would think however that it depends to some extent on how long the oral tradition lasted and what other written sources there were, and which of those were used as sources for the Gospels that we see now.
This actually something that make me really suspicious - there is frankly too much detail for this to have started off by oral tradition.
A good example is that the gospels are stuffed with claimed direct quotes - Jesus said this, Herod said that, Peter said the other, an angel said something else - all with claimed direct quotes. Who on earth was recording these direct quotes - this isn't something possible under oral tradition. No-one remembers word for word what is said in a conversation, although they might remember the general gist of the conversation. Yet we are led to believe that we have all these direct quotes. Now the only way this is plausible would be if someone were present diligently recording everything that was said. Who was this, noting that person would need to be present at times when the narrative suggests there were just two people alone together.
It simply beggars belief that this is possible, leading to the only plausible conclusion - that the direct quotes are made up. Jesus never actually said this, Herod never actually said that, Peter never actually said the other, an angel never actually said something else. These are all made up for narrative effect by later authors. And if these direct quotes are made up, why not all sorts of other claimed details (most of which can never be verified anyhow).
The level of detail seems to me to be clear evidence that the gospels are largely fictional narratives, even if based on real people and events that may have some basis in fact.