I feel that the premises of your question are somewhat unrealistic.
1. Firstly, radiation fades very quickly. https://emilms.fema.gov/IS3/FEMA_IS/is03/REM0504050.htm In that source, it says "for every 7-fold increase in time after detonation, there is a 10-fold decrease in the exposure rate." In a few weeks or months after the detonations, survivors will be able to leave their bunkers without too much danger. Cancer rates might be up for some years, but most people would be fine.
2. In general, you do not go to space to flee Earth, especially due to radiation. No matter how devastated Earth got, space is probably more hostile. In space, there is lots of radiation, along with no air, few resources in interstellar space, etc. If you have the technology to build a ship the size of New York state that can sail the void for thousands of years, you could also build underground, self-sufficient arcologies on Earth that could save a much larger portion of the population.
3. There is no way a spacefaring civilization can lose to a non-spacefaring one. Numbers just do not matter. Any spacecraft capable of travelling between stars will carry incredible amounts of power (Don't have exact numbers, but even a modestly sized ship the size of an aircraft carrier would need energy equal to over ten times the entire modern world annual's energy budget to go up to 10% of light speed). A ship the size of New York could melt the surface off Mars with its engines alone, or just fling a couple of asteroids at it fast enough to smash the crust to pieces, even if it had no weapons, and it would probably have at the very least some lasers to shoot debris with to prevent hypervelocity collisions from destroying it.
4. Even if you are fleeing earth, there is no reason to immediately head to another galaxy. If you can build or somehow found and can maintain a ship that lasts millennia, then you have no more need for planets. Even if a hypervelocity black hole flies through the solar system and eats the Earth or drags it into the void, you could just relocate to the asteroid belt or the moons of Jupiter or something. 947
There are tons of resources there, and you could just build habitats such as O'Neill cylinders and mine the asteroids for materials to repair and expand with. You might eventually want to move out to colonize the stars, but you would only do that after you had established an industrial base and constructed a partial <PERSON> swarm or something like that. You gain nothing by wandering the stars immediately, as the void between them is hostile and largely bereft of resources.
5. Even if you left the solar system, you would not immediately go to another galaxy. There are many stars reasonably close to earth (according to wikipedia, 52 within 16.3 light years), while the closest galaxy is 25,000 light years away. To reach any galaxy, you would have to fly for hundreds of thousands or millions of years through empty intergalactic space, where there are almost no resources for resupplying and repairing, while to go from one star to another, you can just hop from one star system to another, resupplying as you go, with each trip taking just a few decades or a few centuries at the most. Also, there are many chunks of ice and rock, as well as rogue planets within the galaxy to resupply from, though slowing down to grab one and speeding up again might not be worth it. FTL would not change much about this. If it just speeds up travel, letting you travel at 10 or 100 times the speed of light, it would still take orders of magnitude more time to leave the galaxy than to reach a nearby star for no benefit, while if you had some sort of teleportation style FTL, you might as well just teleport to some perfect location, and you wouldn't need to spend millennia in the void.
6. Humans would find it almost impossible to survive on alien worlds with mesolithic technology. Even if they have life of their own, the conditions would probably be totally wrong. The life there will almost certainly have radically different biochemistry, and will be either indigestible or poisonous. We will not be able to grow our own crops in the soil, which will probably be incompatible to their biology. The local viruses will probably not be able to infect us, as viruses are highly specific in their hosts and can only enter the cells of specific organisms, but their bacteria and eukaryotic pathogens may be able to grow in us, and we will have no defense against them. The atmospheric composition will probably be different, making it hard or impossible for us to breathe. Life bearing worlds will probably be almost as hard or even harder than barren rocks to colonize, so their is little point in searching them out.
7. 99