I don't think the Reapers will be in any hurry to spread beyond Earth in a hurry, remember that it took them centuries to methodically wipe out the Protheans...of course back then they had control of the Citadel and the relay networks and could afford to pick the systems off one at a time...
We don't yet know how the Reapers got back into the galaxy, so it's impossible to say what their effect on the relay network will be. I read an interesting idea that if the Reapers did manage to deactivate the relay network so only they could use it then the Normandy would be the only ship capable of traversing the galaxy as it has the Reaper IFF. That seems like a cool idea, and it would give you a reason for travelling around the galaxy looking for help as everyone would be isolated and nobody would know what's going on. Otherwise, one would imagine that the council races would be forced by law to help Earth in its hour of need and you wouldn't need to go around the place convincing people.
Anyway, a Bioware rep on the ME forums confirmed that there will be more ME2 DLC released to bridge the gap between the games, so there's a small chance we may get some answers about how the Reapers entered the galaxy in those.