Looking through the codex I think I found an answer to the problem I had:
The Reapers are thought to be capable of traveling 30 light-years per day, which is 10,950 times the speed of light, and don't seem to require fuel or a place to discharge their drive cores.
The same entry says that Citadel ships top out at about half that, meaning they can make the 2.5 million light-year trip in about 500 years. So I guess the issue about going fast enough to make the trip in an amount of time that still counts as "centuries" is solved regardless of whether they manage to attain the speeds Reapers could.
If they can figure out how to apply the Reapers' lack of fuel or discharge requirements to their ships (which is probably what will happen) then they can make the trip in centuries.
Alternatively: There totally is a mass relay to Andromeda. It just kind of went the way of the Charon Relay. It's rediscovered during the post-war clean up and (after probes are sent through to make sure it's not like the Omega 4 Relay and launches you into the accretion disk of a supermassive black hole) they send a team through to explore.