From what I have pieced together from a number of different sources over recent times is seems that the UK does indeed have an emergency broadcast network. It consists of selected FM transmission sites that have backup generators that normally carry the BBC county radio stations.It is not the national FM networks as you might expect but combined coverage would still be extensive (and power consumption lower).

For the mobile phone (cellular) emergency network, there is a network of 4G sites with backup generators to give coverage of most of the country. It is not clear if these would be accessible to anybody or just selected users.
The YouTube channel, 'Ringway Manchester' made a video about this cellular network only recently.

There is no indication that AM sites would be brought into use as few people would consider going there to check for broadcasts but you have to say never say never I guess if needs must and a situation persisted for a long time.