Probably sooner than that. The "Pope" as we know it is an office that evolved later, though the Catholic church claims the line goes back to the Apostle Peter.
The stories in the gospels are a collection of stories that had been circulating orally among the first century Christians, and got written down mostly in the late second half of that century. Mary likely never claimed a virgin birth at all; that was invented by the oral tradition. Pretty much everything about Jesus childhood is made up to push certain religious narratives.
Which itself is sometimes interesting to follow. The whole census story behind Jesus birth, for instance, is almost certainly made up. Why would Rome require everyone to go back to their birth city to register? That's hugely disruptive to everyday life if people have to travel days or weeks just to fill out some paperwork. But why did they stick that in there? One good answer is that the particular group who wrote that section of the gospels--it doesn't appear in all of them--really wanted to connect Jesus to King David and Bethlehem, but everyone knows Jesus is from Nazareth. So they stick this convoluted census story in there to have a reason for Jesus to be born in Bethlehem instead of Nazareth.