If the pregnancy is "pre-existing", ie: before joining EK, and even (I think I recall) within 6 months of joining (maybe 12?), you are responsible for ALL medical expenses. This also includes any birth abnormalities or genetic problems for the life of your child!
As for Nationality it will be up to you and your wife to register the birth (if born in the UAE) with your home country and obtain a passport.
You will never have a UAE passport!
My advice, until the birth, is to maintain your health insurance in your home country. The EK clinic and medical support is only designed as a minimum cost model to keep your family here, and you flying until you can't do it any longer.