why 10 years at all? a black hole can't be accelerated to even a high percentage of the speed of light, unless you use magic. a black hole can accelerate matter to high speeds, but this matter leaves the maelstrom as simple atoms and ions at most. The next black hole large enough to accelerate a black hole to higher speeds is 25000 light years away and this black hole doesn't move measureable distances relative to the center of mass of our galaxy, because it is essentially the center of mass. So, no gravity assist.
Even if this black hole would then be travelling at 99% c and stay intact, we would have centuries to detect its effects on other stars or dust clouds between us and the center of the galaxy before it reaches us.
So, the scenario itself is already pretty unrealistic. There would be far more realistic dangers to us... what if a Wolf-Rayet star in our neighbourhood goes Supernova? There are some candidates, that are literally aiming their guns at us. While it will very likely not cause instant death to all, because the atmosphere would ionize and form a shield before radiation would be fatal for us, it could cause massive climate changes.
We can't even predict the collapse of such a star currently, because we don't know enough about the collapse of stars. It would be surprising - when we see the supernova, the radiation has also reached us.
So... the reality of the question is: If we don't start making space our home now, we will maybe never have even the chance to evade such a lightning from the blue scenario. It could happen every minute and we might be unprepared.