You wouldn’t actually notice anything. When a large object is far away, all the light coming from it is concentrated into a smaller area than when it is up close. This means that the actual surface brightness of the object doesn’t change as it gets closer: the light appears brighter due to the closer distance, but spread out over a larger area. The Andromeda galaxy is already spread over an area of the sky wider than the full moon, and you’ve never noticed it; only the core is visible as a very faint star which turns fuzzy when you look at it in a telescope. You will remain essentially invisible for most of the rest of that 5 billion years. The shape of the visible portions of the Milky Way may slowly change over the last few hundred million years (and for long after), but if you’re nearsighted or live in a city then you’ve probably never seen that either. Obviously even if you could see them, the changes would all be spread out over such a long period that you would have trouble noticing anything. You can find some sped–up simulations on Youtube though.
Yeah, I'd love to have a night sky that looked like the 4th image in that sequence. I love long exposure photography of our current sky, but to be able to go out and capture something like that would be amazing!