It must be possible to do at least some JAA IR training in the USA, because some airlines have been doing it for years.
The other way to do this is to get the FAA IR and then follow the conversion route to JAA. In Switzerland you can convert with just the checkride. Plus the dreaded JAA IR ground school of course; no way to avoid that
In the UK you can convert with 15hrs dual time, checkride and ground school. There is some outfit in Norway which offers some interesting options, too. This way, you end up with both FAA and JAA IRs (and hopefully with standalone PPLs/CPLs from both) and you can fly not only Euro-reg but also N-reg, fully IFR, worldwide. In this "paper collection game" you can never cover too many bases