Which specific version of English are international pilots and ATC required/recommended to learn?
The FAA bases English proficiency on the ICAO Level 4 English Language Proficiency Criteria, which don't mention any particular dialect or variant of English. Instead, they focus on the speaker's ability to communicate and they explicitly accept that the speaker may make mistakes (emphasis mine):


  Pronunciation, stress, rhythm, and intonation are influenced by the
  applicant’s first language, but only sometimes interfere with ease
  of understanding.
  [...]
  Basic grammatical structures and sentence patterns are used creatively
  and are usually well controlled by the applicant. Errors may
  occur, particularly in unusual or unexpected circumstances, but
  rarely interfere with meaning.
  [...]
  There may be occasional loss of fluency [...] but this does not
  prevent effective communication


It's clear that ICAO doesn't expect non-native speakers to be fully fluent, and once you accept that then there's no benefit at all in trying to enforce a particular dialect. Using standard vocabulary and phrases is the real key to effective technical communication, and you can see that in many other professional contexts too.

I've met plenty of non-English speakers who can speak effectively but not fluently about a specific technical domain, but they still don't have the vocabulary to talk about their holiday plans or other 'trivial' topics. Asking what dialect of English those people speak is fairly meaningless, but Globish might be the best answer :-)