When do we leave:
Mac Walters stated in an interview that is is after the original ME2 Story in 2185.... and abvoiusly before the Arrival DLC.
Link:
youtube.com/watch?v=B-Ef8FD_GU4
The Reapers invade the Milky Way around 21th of Septembre 2186. Minus 6 months for the effects of Arrival DLC (If we didn't, the Alliance will do it) gets you to March 2186....
My bet would be around November 2185 as the launch date.
Why was it build:
The Andromeda Initiative and therefore the Idea for it all was instituted in 2175. Adding ressource gathering and research on top, you can assume the construction began somewhat around 2180 for the Nexus, as it is quite bigger than an Arc.
Pretty sure Sovereigns attack speeded things ups, especially if you remember that in ME2's news broadcasts the damage to the fleets was not completely repaired, but in 2186 the Fleets were raised by a great margin (just look how many ships the Alliance had in the Intro of ME3).
Building them in that short timespan must mean they have huge shipyard capabilities. And the Arrival DLC will have set said things in motion to be at least a bit prepared. Garrus was given a position as a Reaper Expert aswell, remember? But this all happened after the Launch of the Nexus and the Arcs in late 2185.
How do we travel:
FTL speed only. No Relays here, otherwise the Reapers would visit and harvest Andromeda too, wouldn't they?
Because the travel at 12-13 ly a day or so it takes 600 years to get there (simple math: 2.500.000 light years / 600 years / 365,25 standard solar days = 11,4 ly/day).
The engine is the same as we already have, only the way to discharge the core is new and tailored to long range flights.
Shep refuses and the Arcs return:
They will find cleansed Milky Way and a cleaned Citadel but no more Reapers, as they will have finished their harvest by then.
Traveling to Andromeda and back takes 1200 years.
Almost all warships our cycle has left where at earth and by that time all major planets and colonies were already under attack.
We are not as advanced as The Prothies were. If Shep refuses, all notably military power will be erradicated within view hours past the decission.
After crushing this resistance they'll switch to standard harvest pattern mode. The Protheans were harvested in some Centuries. I give us 300 years. Afterwards it is time to clean the mess up so the next cycle would be as clueless as before.
As the Arcs return to pick up more Colonists, there will only be the standard crews aboard. So they arrive, find nothing and be the first to 'discover' the Citadel..... they are like damn... what happend here... we have to get the people back here..... another trip to Andromeda and Back, now filled with Colonists to repopulate the Milky Way and bang.... The Yahg as the now dominant spacefaring and Citadel owning species is far too nice and blows them up. Remember it's now 2.400 years later and the Yahg already know that life beyond there planet exists. The Reapers spared them so they will be the next Cycles race and we are far too few (80.000 induviduals in the Arcs vs. Millions of trucksized Yahg).
There is a plan to estabilsh some safe and fast route between the galaxies for sure, but even we will never live to see it comming, except maybe there is a super high tech device for us to find in Andromeda that makes it possible to bend Space and Time. Otherweise traveling is always acceleration and deceleration and that takes time.