The National Geographic documentary "Evacuate Earth" gave us a doomsday scenario of a Neutron star on its course to hit Earth in 75 years. Even with such an advance to realize ONE generational ship, escapees made it at the last moment, what between riots, defections, suicides, increasing catastrophes, etc.
10 years? In an even more hurry. Orion cannot be built on Earth, you must do it into space. which means USA, Russia, China, EU, India, Japan, Australia, will have to launch and launch and launch materials, keep their austronauts working to an extremely stressful clock, letting them understand that perhaps not all of them or their families will leave Earth.
Riots and wars will spread to get a ticket to safety. Mass exterminations will occur. The candidates will have to be protected 24/7, that is, unless the very guards break down and take the candidates as hostages to get a ticket themselves.
Terrorists and religious zealots will have no trouble convincing hysterical masses to sabotage the efforts by cutting lines of resupplments to the project.
And not to mention plain accidents. You can't do all of this, even in the best conditions and not expect accidents to interfere with the schedule. Two Space Shuttles were destroyed because of...what? A tiny leak in the components, a crack here, and boom! you lose a crew of specialists and a reusable transportation. And you MUST reuse as long as possible since the industries in the world will be under catastrophic pressure to build components for the Ark.
Let's just take the weather: window launches for rockets requires certain meterological conditions, and you can't just come up with innuendo launch stations. Delays will be unavoidable even on that front.
And the more time passes, the more pressure is exerted on the staff and the crew. Medical staff will skip one simple routine procedure with their astronauts, sicknesses set in, more losses. The candidates will start panicking. They are essentially prisoners, they cannot leave the evacuation point's premises -in fact, it would be dangerous even to let them walk around unguarded. Many one candidates will get nervous brackdown or just commit suicide or refuse to leave in order to stay home as doubts creep in deeper and deeper.
Nah, 10 years is not enough.
EDIT AS PER MISTAKEFULLY READING THE QUESTION:
Since the Ark is a gigantic, untested prototype, and such a complex titan that is supposed to host human survivors, seeds, DNA from at least farm animals and enough human DNA to grant reproduction without inbreeding, is likely to suffer system crashes during its travel. Nuclear propulsion, the only more reliable propulsion technology (and energy generations for the ship's needs) at this point of the story, would use 1,000 years only to reach Proxima Centauri B.
My suggestion is that that convenient planet is located around that sun. It's difficult to imagine the survival of the species in such conditions beyond this limit. In fact, in 1,000 years, arriving to Proxima would be nothing short of a miracle