In which case the USN would come into direct confrontation with the MN too. Though in my version of your timeline there isn't a Suez War or not one as we know it.
A very difficult situation, indeed, but it shows the very wide range of possible scenarios.
I think I got the difference between your timeline and mine. From the storytelling perspective, I want Britain to suffer the post-war fall from grace and feel it, because otherwise most of post-war material, such as books on naval design, etc, becomes useless, as situation is very different. And only then re-emerge, kind of like today's China starting from junior partner and working up. That way it is possible to use as many examples from real history as possible, mirroring events and technologies if needed.
Switching too early, as you suggest, with most of the Navy intact, with post-war victorious position, indeed brings to the table a totally different strategy, and totally different mode of cooperation, probably much more aggressive and at least much more assertive, both for Britain within the bloc (being not the second power, but one of two first powers, making the alliance practically equal) and inherently the bloc on the whole.
Some non-naval things to consider.
This requires a very detailed answer, and there is a question of PRC, maoism, as well as Yugoslavia and the whole uncommitted movement.
And Gibraltar. There is Gibraltar. Probably with locally deployed nukes, just in case.
(pro-Soviet communist UK)
The basic assumption is not communist Britain, but a socialist monarchy within a pro-soviet military alliance. One bloc-two systems, that kind of thing. Not a critical factor, but I think it's still important to take it into account. But everything else is on point.
dissuading American involvement in "European affairs"
As counterweight, there will be much more active French participation in NATO affairs. France will feel threatened (and rightly so), as well as West Germany.
I'd expect some surface warships with heavy AShM batteries similar to Soviet designs.
I kept the missiles on the carriers (20 Granits) and on command cruisers (10 Basalt), because it felt a natural thing to do under the circumstances. There are never enough of those things.
I'd expect the US to openly support the cause of the IRA and aid it in being as much of a headache to the Red UK as possible.
Oh, yes, for sure. Ireland is going to be a perpetual sore point, as i noted above, including crazy missions of American SSNs for smuggling weapons.
The UK's navy would naturally become a lot less focused on protecting convoys and instead be more "offensively minded" in the sense of being able to interdict any movement across the Atlantic.
Also, I tentatively suppose that the Soviets would delegate the responsibility for all British colonies, former or current, that still remain under the bloc's control, back to Britain (because it's the natural thing to do) so there will be actually an active colonization under rhethoric of decolonization, and that's what I referenced when I wrote about the destination of the Escort Cruisers as colonial control ships. That will reverse the real "withdrawal from East of Suez commitments".
I wonder what else would US change in the doctrine and overall approach, especially in amphibious operations, as they have to make do without Harriers.
And yes, let's keep the lid on thermonuclear apocalypse, and consider even hot war events as limited and conventional