I did the license conversion. First off, JAA does not exist anymore. It is now EASA.
Secondly, you need the EASA class 1 medical. The country which holds your medical records is the country that is responsible for issuing your medical. I did the Class 1 in the UK and cost me about $750. When you renew your medical each year, it will cost you closer to $150 to renew each time. The EASA medical is more stringent than the FAA class 1, so if something comes up during the EASA medical that you were unaware of, it could affect your FAA class 1. This happened to a friend of mine and the EASA authorities are requiring him to get heart surgery, and subsequently the FAA will require him to get it as well now (for an asymptomatic condition that he was previously unaware of). Initial class 1 exams can not be done in the US. Renewals and revalidadtions can be done all over the world by approved AMEs, but you will have to make the trip to get the initial class 1 done.
Thirdly, you need to complete the ATPL theory exams. There are 14 exams, some are really easy, and some dig so deeply into subjects that you would never need to use that information for the rest of your life. It took me 6 months of nonstop studying. (I did distance learning). Since you already hold an FAA ATP, you will not have any requirement to take a formal ground school. You can self certify for the exams, but you definitely have to look at the question banks in depth before attempting the exams. The ground school that I used told me that one FAA ATP holder purchased their materials but did not even look at the question banks, and when he took the exams, he failed all and scored about 50% on each exam. Radiotelephony is even different by a lot (especially in the UK), so you have to look at those questions as well. Exams can be done in Florida once a month. There are a handful of EASA schools in Florida, but I would avoid certain ones such as NAC and Pan Am. Another Florida EASA school just shut their doors after failing an inspection recently. You will have better luck with Bristol Ground School, Oxford, Gulf Coast Training Solutions, and CATS. There are several really good question banks available for subscription. There are approximately 15,000 questions in total.
Once that is all said and done, all you need to do is find a simulator of the aircraft in which you are type rated and get an examiner out there to the sim. Easier to do if you find a sim in Europe.
You will need to do a formal exam for a radiotelephony license. This is done on a computer. You wear headsets and an examiner has a headset in another room and plays ATC. This is relatively low threat, but some examiners fail applicants in order to get a retest fee.
You will also need to have an English Proficiency level evaluated. In the UK, you can get a level 6 automatically if your examiner is also a level 6 English speaker. Otherwise, if you go to another country, you will actually have to pay for a formal English level assessment. A UK radiotelephony examiner can also sign you off for level 6 English during your radiotelephony exam.
All in all, costs vary widely from country to country, but you can expect to spend around $10,000 when you are all said and done.