You are quite correct, it doesn't.
ANO Article 21(3) states:
Subject as aforesaid, a person shall not act as as a member of the flight crew required by or under this Order to be carried in an aircraft registered in a country other than the United Kingdom unless:
(a) in the case of an aircraft flying for the purpose of public transport or aerial work, he is the holder of an appropriate licence granted or rendered valid under the law of the country in which the aircraft is registered or the State of the operator....
Instructing is aerial work and, therefore, you need a licence and rating granted or rendered valid in the USA ('State of the operator' refers to Public Transport). What you have is a piece of card granting you Private Pilot privileges on the strength of your UK licence, whatever it may be, and subject to any limitations on that licence (e.g. as or with co-pilot, etc.). To instruct on an 'N' registered aircraft you would need to hold FAA Commercial, Instrument and Instructor cerificates.
Also, to carry out aerial work in a foreign registered aircraft in UK airspace you need permission from the DETR.