The possibilities of two ships bumping into each other in the vast infinite reaches of dark space are cosmologically smaller still. The possibilities of two ships bumping into each other within the same galaxy are astronomically small. Its a quintillion times even more true in the vast empty swaths of dark space between galaxies. mindbogglingly, incomprehensibly, inconceivably, unimaginably big.
All the Arks have to do is leave via somewhere like Ismar, Shrike, or Titan and they're well away before anything kicks off.
As for bumping into them in deep space: They come in through the Viper Nebula, from which it stands to reason that they sleep out past Viper.
By the time the Reapers enter the MW in 3 the Arks would be a ridiculously tiny speck of heat against the CMB (to the point of being literally undetectable to all but the best equipment ever built in-verse!) and the Reapers have bigger fish to fry. They're not going to bother looking for a sleeper ship bound for another galaxy. In 3 they only rumble the Normandy when the plot demands it or when you use the scanner, and they're actively seeking Shepard at this point because it's gotten personal.If Bioware went so far out of its way to establish this game in a new galaxy to get around the ending of ME3 and make this an isolated story, it is highly unlikely the ending will have any effect at all. If synthesis was chosen, the saboteurs will attempt to cyborgify the Andromeda Galaxy, and it is up to you to decide whether to permit them or stop them. If control was chosen, Shepard ordered them to go back to cryo sleep and continue their mission. If destroy was picked, the saboteurs became gibbering catatonic basket cases who then starved to death. However, the activation of the Crucible changed things. Alternatively, some people on the Ark ships are probably indoctrinated and intended to sabotage the ships, by waking from cryo sleep very early.So, if the expedition stayed within Council Space, perhaps via the Nimbus Cluster or Silean Nebula, they can get to the intergalactic void unimpeded. At that point the Reapers were all clustered around Batarian space, intending to steamroll them on the way to Earth. They left shortly after the Bahak system was destroyed by either Shepard or the 103 Marine Division.So in the infinitely small (and this one means that literally) chance that the Andromeda Initiative does encounter the Reapers during said journey, they're probably not going to aware of their presence anyway, and vice versa. The people on the expedition are already in stasis for the long journey to Andromeda by the time they enter dark space, and as for the Reapers they go into hibernation after every cycle to conserve their power. Also, there is the factor of hibernation.Space is big, and the ending of Mass Effect 2 implied that the Reaper fleet stayed huddled pretty close together.A minor one and the game has not been released yet, but: It's confirmed that Andromeda project took off sometime between Mass Effect 2 and 3, so how did they left the Milky Way without the risk of running headfirst into the Reaper fleet? Consider that in ME3 the Reapers are capable of tracking down and destroying the Normandy (which has stealth devices) inside a space the size of a solar system.