My theory is that construction of the Ark started after the events of ME1, progressed thru ME2, and became a top priority due to Sheperds actions in the Bahak system during Arrival, when Sheperd managed to stall the Reaper invasion, & the Ark was completed & launched at the start of ME3 whilst Sheperd was in lock down.
As for what prompted them to start on the Ark so early, well if you remember rightly an awful lot of Soverign was unaccounted for & that was blamed on scavengers, I think the council races ( Turians/Asari/Salarians ) spirited away the wreckage, studied it, found proof in Soverigns data core that confirmed what Sheperd had told them about an impending Reaper invasion was true, realising they couldn't beat such a numerous & formidable foe they came up with the Ark plan.