Both France and the UK have floated the idea of extending their nuclear umbrella to include other European nations. Both countries operate ICBM carrying submarines so don't have to deploy nukes on anyone's territory.
I wouldn't be surprised if it happens on the fact they could share the cost between nations alone for something they've already paid for.
The UK’s nukes are sourced from the US. If they want to be independent, they’ll have to develop their own.
For other countries, the problem remains that if the UK or France government turns far-right, the other countries may quickly be on their own again, just like what they now fear with the US.
An EU-level control over the nukes seems unlikely, as France (and even more the UK) want to retain their sovereignty over that.
The UK's warheads are UK made by the UK Atomic Weapons Establishment (AWE).
The UK lacks an independant warhead delivery system since the cancellation of the Blue Streak in 1960, and currently uses US missiles for the UK submarines.
The UK was the first country to start a nuclear weapons programme and talked the US into starting the Manhatten project.
Once subsumed by the US project, the UK started again post WWII and developed warheads and bombs tested in Australia. They managed to dust a future UK prime minister with fallout as a child in Adelaide.
I wouldn't be surprised if it happens on the fact they could share the cost between nations alone for something they've already paid for.