I moved here 2 months ago myself from UK and had to face the same problem as you. Most of the good schools are already full. The worst affected school group is year 1/2.

Ask your firm if they have special arrangements with certain schools. Ask your HR coordinator about their preferred schools. I am sure they can get your child in a school.

Having failed to find a slot at any school, I barely managed to enroll my daughter at 'Newton International School'. They have a new campus built in 'Barwa City' and are open to admissions.

http://newtoninternationalschool.edu.qa/

Feel free to pm/ask should you need help with anything else.