This is an interesting scenario - AM radio being used as an emergency broadcast network in times of national emergency - natural or manmade (war). There's little doubt that good old AM radio can reach the parts other platforms would not reach even with limited transmitters and power. The receivers 'just work' and do not rely on any additional backbone.
As long as you can get a programme feed to a broadcast site, it can be transmitted to the population.
On the negative side, I have to think that a very high percentage of the population would not even think to revert to AM radio to even check if there was any emergency broadcasts on there so it would not have the desired effect. Most would be waving their mobile phones around in the air trying to get a signal and if that did not find anything would just give up and roam around aimlessly.
A good few transmitter sites that carry BBC local radio have backup generators - they are all FM and DAB though these days (apart from about 6 currently) so only cover a local area.
What robust programme feed systems are in place I do not know. Maybe somebody on here does know for sure.