AIUI, the UK CAA issued an AAN in various circumstances, one of them being to cover an install of something that had an FAA STC.
There is an FAA-CAA mutual STC recognition treaty but it does not provide for
automatic FAA STC acceptance. (I believe Australia does automatic FAA STC acceptance).
Many European countries have such treaties with the FAA but EASA (I wrote to them a while ago) pretends they don't exist. You can find a list of them on the FAA website...
AIUI, the CAA stopped issuing AANs when EASA took over certification.
If an item can be found in the AAN database, then it can be installed in a G-reg and by extension in any EASA-reg plane.
However many G-reg installs do not have an AAN, because they were done under the mistaken but widespread belief that FAA STCs can be installed directly, due to the FAA-CAA treaty.
I would not install an item in a UK based N-reg which has no chance of being EASA approved. This is not because I believe that EASA will kick out N-reg planes (I am sure they won't, and neither will they be able to subject them to EASA maintenance regimes because in many cases that would ground them) but because the resale value of a SE piston plane will be impacted if it cannot be transferred to a European registry.
If yours is a G-reg then you have to get EASA approval (unless there is an install done elsewhere in Europe - EI might be able to tell you if they sold any to European non-N-reg people, and you start the search that way). I've got some contacts who can give you a frank and honest input
One would hope the FAA STC would be useful as Approved Data in this case and I am sure that is true, but EASA is European and Europe is morally and intellectually superior to the USA and this must be absolutely respected