Not just small European countries, but all European countries that do not have their own nukes, which is all except France. The issue is, they’ll have to leave the Non-Proliferation Treaty for that (except for the UK, I think), and once an otherwise respected country does that, the floodgates would be open in the world. The other problem is that such a decision would be very divisive in the European country’s electorate, and therefore highly problematic on its domestic political front. This is simply not likely to happen.
A more realistic outcome is that French nukes will be stationed in other European countries. But France is also not willing to give up exclusive control over those nukes, and the next French government could very well be far-right, and thus become as unreliable as the current US government. It’s a difficult situation.
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.
A more realistic outcome is that French nukes will be stationed in other European countries. But France is also not willing to give up exclusive control over those nukes, and the next French government could very well be far-right, and thus become as unreliable as the current US government. It’s a difficult situation.