When exactly did the andromeda initiative begin construction?

The ODSY drive is written to have its primary improvement be specifically about recycling energy for the sake of long-term space voyage. The sort of thing that isn't necessarily so hard for the Milky Way people to develop, they just needed a reason and capital put towards it. And even then, the drives are only so good enough (at least for the shortest possible trips) for the Arks and a basic structure of the Nexus.

ODSY drives are not developed because they have the mass relays and reason to not subject their civilizations again to unexplored systems that could have dangers that will use their relays. The Council (at least not initially and not overtly) is not supporting the Initiative for this reason as well.

It is not faster than the Reapers. In fact, I think I saw something where the Reapers might have even been able to go to Helius and back to the Milky Way in the same span it took for the Arks to get there.

ODSY means the Milky Way people can cross dark space. That's basically the only story-relevant improvement. It doesn't mean it is done as fast as the Reapers, as capably as the Reapers, or that its something all the Milky Wayers could fit onto their ships during or shorly before the war and escape the Reapers. It is something that a recovered Milky Way could use (along with any other future developments; Reapers assisting or not) to themselves go to Andromeda and beyond.

They don't use the existing ODSY drives in the Ai to go to other clusters/galaxies because as capable as the drives are, such extended journeys are likely not what all other hardware is intended for. Its already centuries to get to Helius, I can't imagine centuries or millennia more would be so predictably safe.

However, we can easily imagine a post-MEA society, using mix of MW, Remnant, and I guess Kett tech, becoming increasingly very capable of traversing Andromeda, albeit in journeys of months, years, maybe decades, instead of MW's Mass Relay trips of days, weeks. But well, that's the whole somewhat-new 'explorer' emphasis, at least until tech gets even better.

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The Ai is started in 2176. In timeline context, this is after the First Contact War, the start of the Alliance and Cerberus and the secret discovery of the Reapers (so all the basic worldbuilding), but the same time as batarians are a problem, human corporations are growing, and Shepard is in his earlier stint in the military. Its a time where we can believably consider things like:
-human corporations reaching for the stars with an eccentric leader
-earlier but not earliest human eezo research
-earlier but not earliest human AI research

The gist is that the Ai is what an aspect of human capitalist interests (that doesn't seem directly connected to Cerberus agenda to grow as much capital ASAP to prepare for Reapers) is doing while having growing power and influence on the galactic scale.

From 2176 to 2183 (ME1) its several years of trying and either failing or not having enough money to keep pushing. Since ME1 there was an influx of resources from interested parties allowing the already in-progress initiative to finish within a couple more years - or at least some form of 'finished'. Several months or so later, the Reapers arrive. If a Quarian ark arrives, it'll be either from a galaxy that doesn't have Reapers but maybe at least more chatter about them, or from a galaxy invaded by the Reapers but the ark scrambles out of there, or much less likely a galaxy post-war and for some reason the ark still leaves.

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To more directly answer:
-The Initiative was founded years before ME1.
-It was losing momentum months to years before ME1.
-It hugely gained momentum after ME1 and finished months before ME3. Ai final staff prep is the year before ME2 (as in the time of Shepard being dead).
-I'm pretty sure they leave after main ME2, but unknown about possible 'bridging' events of ME2 DLC. In any case, same year as ME2.


I think it can be supposed that if there wasn't imminent Reaper invasion but still the increased resources, the Ai would have left some time, maybe years after ME3's time. And if there wasn't Reaper invasion or increased resources but still a sustained project (though we know this wasn't happening),
it would have been decades after ME3. So the idea is that outside interested parties helped things and raised concern or alarm to get these helped things out ASAP, but the Initiative itself remains a wild dream of someone with entrepreneurial spirit and initial resources to act on their dreams.