The story is set 35,000 years after the year 2024. In 2024, Donald Trump makes fun of Kim Jong Un's stupid hair, and long story short, the Earth gets nuked to the point where everything is covered in radiation and uninhabitable. New Zealand is the only place where humans still survive.
Since New Zealand does not have enough space for the repopulation of Earth, the entire national budget is redirected to space travel. By 2040, they figure out a way for long-distance space travel to work. They are all set to found a colony on Mars, but the mathematicians screwed the trajectory up and now the ship is slowly heading south and out of the Milky Way. A decade or so after the launch, Toba in Indonesia erupts, wiping out the last people on Earth.
This ship can move at around 100x the speed of light, which is astronomically fast compared to anything we have now, but painfully slow compared to ships that are developed later on. Generation after generation plays out on board the ship. It takes around 40 generations until the ship enters a new galaxy, which is dubbed the South Galaxy.
Over the next 35,000 years, the descendants of these colonists would spread out and colonize the South Galaxy. However, they did not bring technology with them (apart from their ship itself, which was destroyed when it crashlanded on the northernmost planet of the South Galaxy) and the people on board the ship were not exactly tech buffs, so society and technology regressed to the Middle Ages. Space travel became possible again when the colonists discovered magic (a story for another time).
Empires rose and fell, great men were born and died, generations came and went, etc. Now for my question: after all this time, how much about Earth is it likely for these castaways to remember?