How do I convert my FAA PPL to the equivalent in the UK / Europe?
As a disclaimer, this answer is based on Googling, not on personal experience, and compared to the US the situation seems very complicated. I'd suggest contacting the CAA directly; EASA delegates licensing to its member states, so the practical details depend on the country you're in. You can read about one person's experience here, although it's in Hungary, not the UK.

Anyway, it looks like there are two processes: validation and conversion. A validation is a temporary license valid for 12 months only (or 24 months, according to some sources), whereas a conversion gives you a full EASA license. The purpose of a validation seems to be to let you fly while you work on conversion; I couldn't find any definitive sources on the process.

As for conversion itself, the requirements I found on various sites are consistent with EASA's own information:


  If you hold a Private Pilot Licence obtained outside Europe or not
  according to the relevant EASA licensing regulations, you can convert
  it to an European one. In order to do this you will need to pass an
  exam on Air Law and another exam on Human Performance. You also need
  to undertake a skill test and have at least 100 hours as a pilot in
  the relevant aircraft category.


That doesn't sound too bad, although I have no idea how much effort and expense there is in practice. If you're instrument rated, though, the requirements seem quite onerous compared to the US because of the large amount of theory required (this is from the same EASA source):


  Competence Based Instrument Rating (CB-IR): This rating is for those
  who already have an Instrument Rating issued outside Europe (e.g.
  FAA). This reduces the requirements of obtaining an Instrument Rating
  to at least 80 hours of theory and 40 hours of instrument time, 25
  hours of which must be training.


And to add more complication, it appears that the UK currently allows FAA license holders to fly using their FAA license but in the UK only, i.e. it isn't an EASA license and therefore not valid across Europe. There's an ongoing discussion with EASA about it, so it might not be a good long-term option but I really have no idea.

There's also something in the UK - and perhaps other countries? - called an NPPL, a National Private Pilot License. But that also has limitations, and isn't valid outside the UK.

Finally, although it's common to see N-registered aircraft in Europe, if you're a European resident then you can't fly them with just an FAA license. You also need an EASA license.

Someone who's actually converted an FAA license to EASA could probably give you a better answer.