As for the UK Trident missiles, it's simply a US-designed missile, built in America and maintained there. Same for the nuclear warheads - so talking of an independent déterrent force is simply a joke.
That's not quite accurate.
The missile is US, but the "physics package" is entirely UK. That includes the fissile material, the design, the assembly, the installation, etc, etc. The re-entry vehicle (RV, with the "physics package" inside) is a UK design based on US specs for mass, aerodynamics, etc. The UK even has its own "penetration aids" inside those RVs. And the most important part: UK Trident missiles are in UK submarines, manned by UK sailors, and under the complete control of UK defense authorities. They are indeed truly independent operationally, which is where it really counts.