Ho hum, very popular question.
In a nutshell....the log book hours count towards any experience requirements of the FAA.
None of the flying licenses are of any use other than for the free issue of a PPL - VFR only.
If you want to use your instrument qualification, you need to take a written exam on USA procedures.
If you want an FAA instrument rating, few hours flying training with an instructor with probably less experience than you, get him to sign off a "fit for test", pass the flight test, pass a computer based written examination on USA procedures.
If you want a commercial pilots licence, few hours flying training with an instructor with probably less experience than you, get him to sign off a "fit for test", pass the flight test, pass a computer based written examination on USA procedures.
If you want an ATP - get your JAA one - then do the computer written and take the flight test....no training needed.
All the above require seperate certificates for single and multi engine - S/E I/R, M/E I/R, S/E Comm., M/E Comm, S/E ATP, M/E ATP.
Don't forget...without the right to work in the USA, any or all of those certificates will be worthless to you - unless you find an N reg aircraft that you can fly in Europe....and even WITH the right to work (Green Card) many companies are now stating "applicant must be U.S. Citizen".