> Europe has its own nuclear forces

That is not really true. Some european countries have nuclear forces. France and the UK.

There are also countries who host American nuclear weapons, but they can't arm them without the authorisation of the US Department of Defense. (Belgium, Germany, Italy, Netherlands, and Turkey)

Europe as an entity does not have its own nuclear force. (It is hard to even know what that would mean.)

As I understand the UK nuclear deterrence is highly dependable on the US, unlike the french.

In what way?

UK Trident missiles are leased from the US. Also, Much of the technology in the UK program is from the US, and the french tech was developed independently.

Russia is in Europe and has a lot of nukes.