Presumably you are a UK FI. In which case you can instruct on an N Reg aircraft in the UK, on your UK licence but not outside the UK. You cannot be remunerated unless you obtain the permission of the DETR to conduct aerial work in a foreign registered aeroplane.
In order to instruct under your FAA licence it would have to be Commercial with a CFI rating. From the sound of it it is Private, the validity is based upon your UK licence, not the privileges.