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Maybe it's the wording but 'the King and his ministers control the Thames Valley west of the ruin that was London'. Charles can not have ministers, and certainly not the PM, in any literal sense The PM isn't even mentioned and IIRC members of the royal family were not to be bunkered with key government ministers. Also, if 'his ministers' is being literal, that puts Charles in charge. If instead it's referring in a deferred way (i.e. when the term 'her majesty's government' is used in a general, not literal, way) then that means something else. RAW it does suggest that Charles is wielding some form of power, in a limited area.

To clear things up, it might be better to reword it and say that Charles has survived, and ministers sharing the bunker at Reading have managed to regain some order in the 'Thames Valley west of the ruin that was London'. I'd also add a line noting that possibly the PM has survived and is in a bunker at X, whilst other surviving bunkers around the country are trying to bring some semblance of organisation and order to their local areas.

The Government is surely Her (or in this instance, "His") Majesty's Government?

Then the ministers surely must, by extension, be Her (His) Majesty's Ministers?

As I've mentioned, that's simply a term - it has no legal meaning. If instead the Alpha means that Charles now literally has ministers etc, that's a different thing.

It is indeed a term, with a historical meaning — and it is also a fairly commonly used one.

I have not seen any reason to interpret its use here as anything else than being within that context. It's the Monarch, and it's His Majesty's Ministers.

If Elizabeth is dead (or assumed so) then Charles is monarch. If the PM survived then fine, he or she has a regular meeting to brief the King on what's happening and get some input (which he or she can then ignore); if the PM is dead (or assumed so) then the various surviving MPs agree amongst themselves who should be PM, they get the nod from the King and then the new PM picks a bunch of MPs to form a cabinet.

Boom! King Charles is in a bunker near Reading and the ministers of his government assert shaky control over the Thames valley west of what is left of London.

I also like the idea of the French invading Britain - not only is it a lot easier for them logistically, but It's possible that they are welcomed by the locals if France is still reasonably intact like they were in the original timeline.

A French humanitarian enclave on the south coast that, by a happy coincidence, has secured the largest onshore oil field in western europe and whatever is still left of Britain's second container port would be an interesting home base for a group of PC troubleshooters.

Good points.
I was trying to ignore the 'The Royal Navy get's swatted aside' in the background, lest I was being to nationalistic / sensitive, but the Soviet Fleet has to get all the way around Scandinavia, and passed Iceland and most of Norway, with pretty much all the RN out to stop them, plus the Norwegian Navy, plus various bits of the European NATO navies assigned to that area, plus large chunks of the USN. Including in that are quite a few very good submarines that would be doing their best to sink anything and everything they could before the Soviets even got clear of Kola.[SNIP]
Someone really hasn't thought this through at all.

Soviet Fleet? No. Satellite targeted MIRV Nukes? Probably.
It won't take more than 2-3 missiles to individually blast the operational groups anywhere near England. And the homeports.

Ouch...

considering is not even clear if the PLA Strategic Rocket Force is capable to it now with their DF-21D... it was even more problematic at the time. First of all even soviet RORSAT Satellites had issues in locating task groups and their information was usually hours late, I think we are talking about 8-10 hours delay. Then task groups sails, thus moves. And in 10 hours at a cruising speed of 20 knots you have plenty of space. Finally MIRV are not accurate against moving targets neither designed to be. They need pre-programmed launch instruction and these requires time. Said that Operation Crossroads show a quite strong resilience of warships to nuclear blasts.

You can take out the ports, but again Underway Replenishment.

Ouch...

considering is not even clear if the PLA Strategic Rocket Force is capable to it now with their DF-21D... it was even more problematic at the time. First of all even soviet RORSAT Satellites had issues in locating task groups and their information was usually hours late, I think we are talking about 8-10 hours delay. Then task groups sails, thus moves. And in 10 hours at a cruising speed of 20 knots you have plenty of space. Finally MIRV are not accurate against moving targets neither designed to be. They need pre-programmed launch instruction and these requires time. Said that Operation Crossroads show a quite strong resilience of warships to nuclear blasts.

You can take out the ports, but again Underway Replenishment.

I think you're not giving soviet intelligence enough credit.
Sure, you might need to work for 4-6 hours to find them, but then it's only minutes to track them over the subsequent passes.
And from fire to arrival is under an; multiple estimates of 15 to 20 minutes for russian ICBMs.
So, maybe 5 nm distance. Quite possibly less, Assuming that they are aware of being targeted and being able to go full steam.

The UK doesn't use nuclear plants, so they all have fuel-oil limits. Taking out the oilers is every bit as important as taking out the surface combatant fleet. They're targets, too.

The UK doesn't use nuclear plants

Um...?

Apart from Calder Hall and Sizewell and Trawsfynydd and what? half a dozen? other power stations running during the 90’s, then...