I think you are wrong in questioning our nuclear commitment to NATO. That was there at least a decade an a half before we joined the EEC, I dont see Brexit changes it a jot. Ultimately if someone starts letting off buckets of Sunshine in Eastern Europe, we are going to be getting the fallout anyway, and the logical conclusion, our being a NATO hub for airfields, communications and so many things, is that we will be next. So we would commit. In fact, even a heavy cyber attack, in theory, would be enough to set it off.
The Americans would be unable to remain uninvolved in the event of a UK nuclear use, because they provide the missiles and some of the targeting systems we use. So they would be implicated however it turned out. There also remains the awkward point that, if we wanted to engineer a US nuclear response, we could arrange one fairly easily. We would just launch our missiles from a submarine parked off the US East coast and allow our enemies to come to their own conclusions.....
Im not sure about the French nuclear capability being nuclear capable. Till it was replaced by Rafale, it was carried by Mirage 2000N's IIRC, which obviously remained incapable of landing on a carrier. I think the Rafales they replaced them with also are land based variants.
Interesting article though, thanks for the link.