Do you have already a valid driving license from U.K ? If you do so I guess that you can go directly and give the street test. My friend is from the Philippines and he got a driving license from there, so they ask him to give the street test directly. So If you have it just show it to them and you are done. What I've understand from your post is that you are willing to learn, so you don't have a driving license with you.

You can try Qatar Modern School, it is near Giant Stores & Villagio. I guess that they will ask you to come after 2 months at most and it is really too much. Being on the waiting list is something common in all the driving schools.

Did you try Al-Raya School? This is a very good one and I do recommend it. The issue again is you have also to wait, unless you know someone there. If you know someone there you would starting in 3 days or a week at most. They will ask you to pay more, but it is better than waiting for ages like you said.

Schools List:
Raya - Freej Cleeb
Gulf - Matar Qadeem
Qatar Modern - Mouayther
Doha School - Khrytyat

Best Regards & Best Luck