I believe the ship leaves before the events of ME3.
I believe that it is a fun narrative to imagine and beats the boredom. A pre ME3 ARK narrative offers a form of closure event, allowing the builder/writer an opportunity to cram all they love about the MEMW into one boat. We already know it is sailing, we already have an image of what it looks like.
Using the applied rules of how technology uplifts worked in the MW major plot movers. (Tech uplifts are achieved through some, unexplained and speculative Black Box process of reverse engineered Alien Phlebetonium) and the timeline of their introductions to the human timeline.
2148/49:- Lore date for the introduction of late Prothean Empire technologies. This technology fuelled the expansion of humans through dozens of colonies, some with massive populations and in exceptionally hostile environments that required technological solutions that were reverse engineered from Prothean technologies discovered on Mars, and in other Prothean sites after 2157. We never saw the apex of the engineering capacity of humans, let alone other species. One multi-species colony has a population of 1.8 billion (Caleston). Established in 2144 by some un-named Alien race, but Eldfell Ashland entered the sector in 2175 and built it into the industrial powerhouse of the Attican Traverse. It is also the centre of drive core engineering for the Alliance. The moon is geologically unstable, highly volcanic and affected by the strong gravitational forces of the parent planet, Cerunnos. Also, has no native agriculture, water and a hostile atmosphere - totally dependent on the engineering of the Bio Domes around the cities, mining, eezo processing, ports and engineering facilities.
2183 Reaper Tech (Sovereign), Citadel and Ilos (Prothean/Inusannon) Little data has been revealed as to what technological discoveries may have been obtained, but there were techs that would be ideal for an ARK (not least the Cryo facility) Another valuable tech uplift introduced was due to the rebuilding of the Citadel after the attack. The materials tech was listed as unknown in ME1, probably due to restrictions on materials testing, but the engineering capacity had been developed in the off screen timeline of Shepard's death
2185 Collector Technologies. This may be totally Reaper Tech derived, or may have some hybrid form with Prothean techs.
It may be a folly to speculate on the possibility of engineering the impossible into some form of narrative, but it is more fun (for me) to re-read the lore for the off screen stuff that Shepard didn't visit.