First of all many congratulations to you!
From personal experience, I would say go to Al-Ahli...
I went to a private doctor during my first trimester and she recommended that I start going to Hamad Womens hospital as I had Gestational diabetes and they have better care there for emergencies, if any....
so I started going there for bi-weekly visits and tests... i had a terrible experience at that hospital... i was just a 'file' there and everytime i went, i was seen by a different doctor, although they said they were from the same team. Most doctors were very rude and impatient..the nurses were good though. The doctors almost always scared me a lot about my glucose level (which was just borderline) and i was stressed before and after each appointment. To see a specialist, you need to go to the evening clinic, which is very very busy and appointments are very difficult..
Finally in my 8th month, i decided to go to al-ahli as the stress of going to Hamad was too much for me. I was very happy with my experience at al-ahli! I chose to see Dr. Sujatha as i had heard a lot of good things about her...she answered all my questions patiently and even gave me her mobile number to call her whenever i had any queries or emergency. I called her when i started my labor pains (in the middle of the night!) and she answered my call and also spoke to the doctor who was on duty at that time in the emergency section.. As i was not scheduled for a C-Section, i just went to the ER and was admitted...i took an epidural as soon as i went in.. my husband was allowed to be with me all the time. The post natal care was also good there. All in all, i was happy i went to al-ahli...
In general, you will receive mixed opinions from everyone about all the hospitals here. Everyone's experiences are different and in the end it will be your opinion that you will have to rely on. So i would say, visit both the hospitals and then decide for yourself which suits you better.... wish you a happy pregnancy!