It's great to see the alpha versions of the various books, but I must say that the section on the UK is still as problematic as it was in earlier editions in terms of accuracy. I know it's just an outline in a basic way, but it completely misses how the emergency/war infrastructure works and how it would shape any post-war environment. This in turn misses out on some opportunities to make the UK an interesting backdrop for games or campaigns in 4E.
The main issue is in two parts: (a) that absolutely no Members of Parliament (MPs) have survived the war, and (b) no local authority leaders have survived. In reality, both parties have a fairly high chance of survival, because they would've been sent to bunkers all over the country when it became apparent that the UK was going to be subject to a concerted nuclear attack. These bunkers are part of a nationwide system designed to manage the war situation as much as possible as it happens, and also to manage the post-war situation. Ministers, MPs or councillors leading these bunkers have full legal powers and are the UK government in every legal sense, but disseminated around the country. This in turn means that they have control over all police and military units in their area of responsibility, and in times of war/post-war have the power to declare martial law - they essentially have the power of life and death over any survivors. It's very unlikely that all of these bunkers would've been targetted by Russian missiles - there just too many of them. Some may have been lost because they were close to attack sites. Enough will have survived to start, in theory, rebuilding the country in the post-war situation. It's quite possible that they are in contact with each other via their own telecommunications network, at least on a local level and possibly with any surviving parts of central government command in their own bunkers. The whole idea behind this is that the country should still be able to function after a war, and then rebuilt. The Army and Home Guard mentioned in the 4E Player's Manual would be controlled by them. If you've ever seen the docu-drama 'Threads', this will give you some idea of how a bunker operates. Similarly, there is a documentary here https://youtu.be/milbW4RDIco?t=1613 that also shows one going through a war simulation. It also features a discussion with the bunker commander about what he would do should martial law come into force - scary stuff.
Because of this, Charles emerging from a bunker near Reading and declaring himself King Charles III is, as far as the UK is concerned in any legal form, completely meaningless. I could emerge from a shelter and declare myself King Jerry I. Someone could emerge from a crater and declare herself Queen Tracy the 56th. None of these claims have any legal basis, and they would not supersede the authority of any remaining government within the aforementioned bunker system. The royal family are assigned bunkers more as a kind of ceremonial courtesy rather than as some kind of legal or decision-making necessity in terms of running a post-war UK, as they have no actual powers in that respect. Aside from that, none of them have any training in such matters. If Charles or his armed followers decided to exercise any 'powers' outside the bunker, the local surviving MP or councillor would be completely within their rights to arrest them. Or put them up against any wall still standing and shoot them. That said, what might make for interesting play WRT this kind of theme is that Charles or whoever else surviving around him declares him king, and tries to set up a power base around Reading. This effectively makes Charles a warlord. The surviving parts of local/national government, who are actually legally the ones in charge, might have other ideas but they might need people with military training to assist their local forces. Or the players might throw their lot in with Charles and his group.
WRT the country in general, if the various bunkers can in fact managed to start organising in the post-war situation - and they are the most able to do so - it's possible that they would need people with special skills and/or equipment to help them. They would be trying to establish some semblance of order above ground in their demarcated areas of control. They also might need people to travel from one node in the system to another in order to establish physical links in terms of the infrastructure, move supplies or personnel. It's possible that they might also have access to aircraft. How dangerous the country actually is after a war is a bit of a moot point. It's more likely to be dangerous in terms of fallout and disease, but armed groups might only be a problem if there are enough deserters around and/or gangs who've somehow managed to access any surviving military facilities. Maybe some of these bunkers have gone rogue - always possible if the chain of command breaks down - and need 'fixing'. Or maybe one has suddenly gone off-line, which needs investigating.
I'd be very interested to know why the Soviets decided to invade the UK, especially after nuking it. I guess it would depend on what the numbers are in terms of their forces and any forces active to counter them. I see there's mention of a battle in Trafalgar Square, but - given the Square's proximity to several nuclear targets within central London - I don't know if the Square would be there any more Anyway, apologies if this is probably surplus to requirements but I thought it might be useful in some way. Scotland and Northern Ireland are situations all in themselves so possibly up for further discussion.
Either way, I'm very glad that 4E is on it's way. I ran 1E from 1984 onwards and ran a game a few years ago and still really enjoy the setting and the rpg