Well, a quick read after a brief absence and a short post before I must rush off again for a bit.
One post mentioned would there be an Imperial FEMA type agency to handle such an effort. I'd say not, it'd be up to the individual world to prepare for any such disasters. The Imperium would get involved when you try to move that population interstellar, more so if the place you're thinking of moving to is in an Imperial system. They may help or deny your attempts to approach.
Off hand, if I did the numbers right (a quick HGv2 design based on evac of an average Pop 9 (5 billion) world. Just double the effort for the 10 billion scenario, the time remains the same.
Just, he says
Depending on the capacity of the world to build TL15 (for the power plant) starships this is a 5 year plan* to evac about 5 billion (5,000,000,000) people and resettle them anywhere within 1 year of travel/time at J1 with wilderness refuelling supported. Note that the capacity to build the ships is the big problem but I'm not aware of that capacity, if it was defined (maybe TCS).
* 4 years if you allow construction time to be adjusted by the same factor used for cost reduction for the hull, a mtu idea I think .
Start with a plan for a 1M ton ship. You will need about 1,550 of them being built simultaneously. Passengers will be processed as the ship is built. They will also form a good part of your labour I guess. Accident and natural death will reduce the population some during the effort and no births will be allowed, simple contraceptives in all food will be mandated to insure this is achieved.
The ship will be a simple streamlined flattened sphere affair, each one built near a suitable source of population and resources. While under construction it's power needs will be supplied by planetary sources. By building on the planet we make it easier to load the evacuees.
After factoring the drives (J1 M1 P1), fuel (1J1 and 4 wks) and purifier, bridge and computer (model/7) and the operational crew's quarters (4,000 crew per ship), life support for 1 year (1 ton per 20 crew per month), the remaining tonnage is devoted to emergency lowberths. No one will be taking much baggage on this trip.
This allows 806,573 emergency lowberths with a total capacity of 3,226,292 sleepers, per ship, in addition to the 4,000 crew memebers. Many of your sleepers will be medical techs who will be the last in and the first out.
Emergency lowberths are ideal since once the people are in they only need to be brought out after a suitable site is found, and even then may be brought out in managable numbers. Lowberths do not consume any resources and are good for centuries though the risks do increase over time.
The one year deadline for finding places for the evacuees is based on the ship's carried stores and need for maintenance. If suitable ports can be found to address these needs the ships could continue to operate for decades.
Anyway that's my quick take on the problem
I agree, it's a fun thought experiment. Feel free to poke holes in my plan, it was rather quickly done, after-all it is an emergency right