As I understand it, the ICAO agreement allows member states to issue flying licenses based on those issued by another member state.
In order to issue a UAE ATPL, the GCAA must have a valid ATPL to transfer. An ATPL is validated by a Class 1 medical. If the ATPL is not valid on the day that the GCAA gets around to processing it, then it simply cannot be done.
Emirates asks that your license / medical combination is valid for at least 3 months after you join, in order to cover any administrative delays in processing it.
I suspect that if you try really, really hard you can find a free hour in which to visit your doctor.