I'm afraid that the view given by Peter is NOT shared by the vast majority of FAA pilots who reside in the UK, for those who hold just an FAA licence it is a time of uncertainty. Pace made some very good points about what is happening in respect to trying to find a workable solution for all of this.
There are many loose ends at the moment with regards the implementation of EASA.
From my point of view I feel uncomfortable with the doubt, even though I hold both FAA and JAR/UK licences. The aircraft I normally fly is N reg (based in the UK and UK VAT paid) but registered and insured through the US and I pay tax both in the UK and the US. I also have both UK and US addresses. Yet even with seemingly all bases covered I still feel unease.
you have absolutely no reason to feel that way.
you are double licensed so you are just fine. it is only about licensing.. there is nothing to prevent a n-reg ac take of into the eu skies.
...
in the us the operator is the one who makes (decides to make) the plane do a flight.
this is not the pilot, and not necessarily the owner (for example a lease company does not decide .. the organisation/person that leases the plane).