Britain's last two territorial conflicts involved the Falklands, and Icelandic fishing ships in the neighborhood of Rockall. I don't recall that third party assistance was required in either dispute, thank you very much.
The US the other hand is always asking for other countries to bail it out of some mess or other.
Did you miss the word "large"? And the UK required significant US assistance during the Falkland Islands war, so apparently you're quite ignorant about history.
The command and control systems are American. The intelligence network is American. The last time the British military acted without US support was Suez. The UK went in on NATO 100%.
France took an entirely different set of conclusions from Suez.
NATO can and probably would shoot them out of the sky using ballistic missile defence. IIRC Britain has a total of 8 missiles.
Britain's obsolete Trident subs are manufactured by Lockheed, so if they fall out with Trump they will lose access to spare parts. The replacements aren't due until 2028.
Are you under some absurd notion that NATO has ballistic missile defense in Russia? Or can magically reach out and intercept a ballistic missile anywhere on the globe?
If there was the right Arleigh Burke right next to the submarine (which there wont be), and the captain was insanely paranoid enough to contravene standing rules of engagement to already be prepared for a surprise ally launching a weapon, there's about a 50% chance they could intercept each missile.