Good luck running such a project when almost everyone you need to pull it off knows that there's no chance that they, or indeed anyone they know, will make it off planet. If you have a couple of years left (which, by the way, is enough lead time to get serious about space launch - by which I mean, drop everything, refocus all industrial production, and let the world burn if we must - and evacuate everyone), would you prefer slaving away so that someone you don't know will survive, or spending them with your loved ones? I know what I'd do - spend them slaving away so that me and my loved ones will survive. Airship to orbit, big dumb boosters, screw the heatshields since we're not going to be returning...
But why would anyone be launching a generation ship? Sun about to go nova several billion years ahead of schedule?