A few years ago, I did exactly that - during a furlough from my airline.
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Obtained an Icelandic (JAR) ATPL and 747 Classic type rating, for which I had presented a current US FAA ATP with 747 type rating, and plenty 747 PIC time. I was a TRI/TRE (747 type rating instructor/examiner) with my employer. So the conversion was a minor excercise. Just see how "they" do it...
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Sat in the class with the other new hires. many of which had no prior type experience, so tried to help here and there. Standard Boeing AOMs, but had to study the differences for RB211 engines I never operated (Mamma-mia, they got N3 on these motors - what's that...?). Just learn the JAR regulations, and pass the JAR 1st Class Medical (British CAA "astronaut level" medical), all of which took some 4 weeks, 1 simulator warm-up and training session, then 1 type rating check ride lasting 2 hours. And finally 2 weeks of line checks with a line-check captain who wanted to re-invent the wheel and correct my American English "gear" to pronounce British Oxford English "undercarriage"... Then all that for, after a month's flying, being recalled by my airline... Oh well, I got to fly the 747-300 and the RR engines... Was fun.
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Depending which airline you go to do this 737 JAR ATPL and type rating, I do not see where it will much different from your FAA qualifications and experience. Suggest you do your JAR 1st Class Medical in a country where the standards are "normal humans"... maybe Netherlands, Belgium or Luxembourg.
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All the best to you -
Happy contrails