Let me run something by you guys real quick. (Patreon)
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So I'm about to have Obama go into detail about how the timeline between ME2 and ME3 looks, and you guys tell me if this makes sense.
Mass Effect Andromeda Initiation (Novel) says rumors of Shepard appearing on Omega surface on Feb. 15, 2185.
Presumably this is when we head to Omega to recruit Mordin, as that's the only thing you're required to do.
The Alliance News Network Report (pictured below) says the Reapers attack the Batarians on Sept. 21st, 2186.

We know the Reapers go from the Batarians straight to Earth, so the beginning of Mass Effect 3 takes place in the same general time as that ANN report.
Shepard is imprisoned on Earth for 6 months, which would mean they turn themselves in sometime in March of 2186.
This is conveniently the same month that Doctor Bryson from the Leviathan DLC talks about the possibility of the Rachni being used by Leviathan. Specifically, Bryson's log is dated March 7th, 2186. Meaning Hackett got Bryson looking for Leviathan either shortly before or after Shepard goes to prison.
Anyway, I figure Mass Effect 2 must take place over the course of around 13 months February 2185 - March 2186. Crudely demonstrated in the Paint timeline below.
This, of course, all assumes Shepard doesn't delay in going back to Earth after the Collectors are defeated and ignores the events of the Arrival DLC.
