France is the only nuclear power in the EU. How does it protect its facilities and maintain the secrecy of technology when the European Parliament has at least some say in French security and military matters. The fear here is of nuclear technology being leaked to non-nuclear states. Do they have any special protections or mandated agreements for this under the EU framework?
The military cooperation in the EU is on the level of relatively vague security guarantees and overall policy coordination. It does not extend to the command and control of military units. EU Battlegroups remain national forces which have been put on alert for possible EU deployments.
There is a lot of bilateral and multilateral cooperation among EU members. often under the NATO umbrella, but that again involves national forces putting individual units under multinational headquarters. France no longer has any nuclear army units, just air force and navy.
I also think that you're under-estimating just how much major industrial nations already know about nuclear weapons technology. The Non-Proliferation Treaty allows full civilian nuclear development against the monitoring of nuclear materials and the promise not to use it for military purposes.