This could mean large numbers of people re-registering in other EASA countries.
With JAA licenses, the registration does not matter.
I know some people do it already but I have never found a clear list of the benefits of being on say a D-reg.
The only thing is the letter of ANO article 140 which bans any aerial work in a foreign reg plane, and while the DfT will give you permission to receive training in one, and some other stuff, this is a hassle. It incidentally also bans the traditional PPL Cost Sharing scheme - the passengers may not contribute a single penny towards the flight. I vaguely recall somebody having got a letter from the DfT saying Art 140 does not apply to EU-reg but have never seen a reference.
If there is an enforced move from N-reg to EU-reg, anybody doing that will naturally shop around for the most appropriate nationality. I should think some new EU members e.g. Romania might provide interesting options. There will be thousands of pilots in Europe doing this - if it happens.