Welcome Andy to GADC
You might want to check out this forum for flying abroad as the rules do differ depending upon the country.
My understanding is you will need to register with Iceland if you have not flown in the EU before. Unfortunately our registrations are not valid in the EU. In Iceland I believe you also have to have your name, address and phone number on your drone.
There is a thread on Iceland here but the regulations have changed since it was started.
This thread is more generic about flying in the EU. You can incidentally register in any EU country and it will be valid throughout.
Enjoy Iceland