I mean; was there ever a mention in the game that the races could travel outside of the galaxy? Or at the very least humans? If the reapers are going to destroy everything in the Milky Way, why stay there? Doesn't the synthesis ending affect all sentient beings?
The way ftl engines work in the Mass Effect game is they build up an electrical charge over time. UNless it is discharged periodically, into a planet or some other massive object, it will eventually discharge into the ship's hull, frying electronics, fusing bulkheads and cooking the crew.
ALso, military ships, at least, have a maximum cruising speed of about 15 light years per day. At such a speed, running constantly, it would take over five hundred years just to reach Andromeda. THis does not take into account time needed to actually find a garden world that can be settled (ie, not occupied by another race, within the proper tolerances for humans, turians, or whatever)
Reaper ships are about twice as fast, and do not seem to require discharge to keep functioning. However, they do build up an electrical charge, as some were reported as being wreathed in lightning when landing on planets during the invasion. This still means it would take over two centuries to cross dark space, and keeping the crew alive is still very much an issue.
So the answer is, no there is no mention in-game of species being able to travel to other galaxies. Not and survive. The Reapers possibly could do it, given enough time, but likely not with live passengers.