There is abcolutely no reason why the UN neds to vacate the building completely during construction. Asbestos abatement is easily done on overtime with minimum disruption. It takes about 48 hours to clear a space completely. And they could reskin the building while the building is fully occupied.
I suspect that part of the problem has to do with each country's security requirements, especially for communications and the integrity of their offices. They can't permit their commo areas to be compromised and there is always the fear of listening devices being implanted in the walls and ceilings. Each country would have to relocate its commo facilities during the asbestos abatement phase, regardless of how long it took. That could still be done on a phased basis. They could relocate to the CG's residence or have their embassy in Washington take over for them in the interim.