The CAA have no responsibility over what happens in Jersey so would not be able to answer your question.
Article 26 of the Air Naviagtion Order 2005 as applied to Jersey by
The Air Navigation (Jersey) Order 2008 states that "A person shall not act as a member of the flight crew of an aircraft unless he is the holder of an appropriate licence granted or rendered valid by the law of the country in which the aircraft is registered or the State of the operator and satisfying the minimum standards established under Annex 1 to the Chicago Convention."
Therefore you should have a FAA licence to fly an N reg in Jersey airspace unless the Director of Civil Aviation has issued an exemption to the requirements of article 26. The reference to "the State of the operator" refers only to aircraft operated under Artricle 83 bis agreements.