I doubt that the speed of light can be exceeded, or that wormholes will be viable. I think the answer will be to travel at .99 c and use time dilation.
All other galaxies are growing old at the same rate that the Milky Way is aging. I don't see any advantage to fleeing the Milky Way. When the Milky Way finally dies, all the rest of the universe will be dead too.
But the Milky Way (or whatever it will be called after it merges with Andromeda) has a very long life ahead of it. There will come a point when all the galaxies in the universe run out of fuel for new stars and the universe goes dark, but our descendants will be able to survive on generated power in a dark universe.
The real death of the Milky Way will come as the result of occasional stars and planets being ejected from the galaxy. It is a very slow process, and there are a lot of stars in the galaxy. But if you run the clock forward countless eons into the future, all the galaxies will slowly boil away as their stars (dark husks at this point) get ejected into the void.
But this is on a massively long timescale. This isn't going to be any kind of a short term problem. We're looking at 100 trillion years before the universe goes dark. After the universe goes dark, we'll have a million times longer than that (100 million trillion years) before all the galaxies boil away.
One further thought: While I don't think there will be any point to fleeing the Milky Way, if we do need to leave we might want to do it sooner rather than later. The accelerating expansion of the universe means that someday all the other galaxies will be receding from us at faster than the speed of light.
But like I said, I'm not sure that there is any point to fleeing the Milky Way. All the other galaxies are aging at the same rate that the Milky Way is.