How do humanitarian flights arrange special permissions to fly when ATC isn't available?
In the U.S. at least, HADR doesn't launch aircraft until a crisis aka disaster has been identified, even if at launch they don't yet know if the problem is lack of potable water or civil unrest over rigged elections.

An already flying commercial airliner won't have a glut of doctors and medicine unless it's been chartered for an already identified crisis, so it's unlikely to divert to this one.

If, after an identified disaster, an aircraft flies there to assist:


A helicopter doesn't need ATC support to land in an off-airport parking lot, of which a major city would have many.
A fixed-wing aircraft, needing an approach path, lacking ATC response, would revert to non-ATC protocols, just like at a small Midwestern grass strip or an Alaskan inlet.  VFR.


Non-aero analogy:  fire trucks and ambulances don't fret over parking tickets.