Be careful
Ladies/Gents
I don't fly for EK, but do have a GCAA licence and medical. I came here last year and fully admitted to having a family history but proved (by letter) that I had been fully investigated by the UK CAA to be fine, no limitations. That was fine for my UAE company CMO/AME and I got the job.
However, 2 months in to the job, the GCAA insisted I undergo similar tests as in the UK. All is fine and I eventually got the licence, albeit after being "medically suspended" for 6 weeks (nothing happens quickly here).
Moral of the story: the EK AMEs might be satisfied with your previous results and offer you the job, but that doesn't necessarily mean you get a licence without further specialist tests. If in doubt, ask about getting the GCAA Class 1 certificate before accepting the job.