Enough warning to build 3 (at least) massive ships, bring the salarians, asari and krogan (at least) on board with the project.
Not to mention develop an entirely new form of ftl travel?
A theory I've seen somewhere says that the colonization ship might have been a previous Council project to explore new galaxies or space sectors with no relay access that was hastily repurposed once the attitudes started to change.
The Galactic governments were in denial about the Reapers up until the time they actually showed up. ME 2 starts only a year before ME 3 and we all remember the "ah yes reapers" speech we got from our favorite councilor. I doubt the council or the alliance was willing to make that kind of investment especially when they didn't even believe in the reapers.
We like to remember the "Ah, yes, Reapers" moment, but afterwards the Collectors and their defeat happened, the Batarian relay incident happened (if it wasn't Shepard, the Alliance sent marines, so that means that other people were aware of the danger) and Shepard was arrested for six months, probably with some interrogation (more information for the higher-ups). The Turians, always the pragmatists, turned Garrus into their official Reaper expert and prepared a strategy in case of Reaper attack which wasn't half-bad.
The galaxy at the beginning of ME3 was different from the galaxy in ME2. The aforementioned six-month arrest provides Bioware a window of opportunity to justify how and why the galactic higher-ups managed to make up a plan B without Shepard knowing it or expecting them to have one.