Steve, a couple things. Yes, switzerland is not part of the EU, however, it is a member of european common market and abides by all basic EU rules including the freedom of movement and the final legal authority of the ECJ. The latter is a prerequisite to be a part of EASA.
No, EASA does not regulate engine approval in Asia or the US, however, many regulators use a shorter certification process that is based on the acceptance of the EASA certification, especially the US as FAA and EASA have a reciprocal acceptance agreement, which will be null and void for UK products come BREXIT if no deal is found, which would in turn invalidate any certification based on that no longer existing EASA certification. Of course everything can be recertified, which will cost a lot of money and some time, as by now not only the initial data is available but also a lot of operational experience and probably some ADs which might change the outcome or certification standard as those have developed over time as well.
Of course the FAA can decide following their own legal advice to still honor the no longer valid EASA certification, but considering the US stance in the open sky negotiations i‘m rather doubtful about that. After all RR competes (not very well seeing the 787 fiasco) with US companies in the same market, and protecting those would certainly fit the America First attitude of the current administration.