I genuinely don't know how they would proceed, but it'd be interesting to watch.
They can physically tap global internet cable just because they can
Member states have spy agencies, but they also signed treaties to join the EU. Having your spy agency violate international treaties isn’t something most governments allow.
If the information is stored in a country, expect that the owner of the information can be compelled to hand it over by a court order (or it can be seized).