Roy

My advice to you is this.
Firstly, get both your children on the waiting list for DESS. People are leaving Doha every week and spaces do become available. Ask where your children are on the list and check regularly so you can (gu)estimate how long it may take to get them enrolled.
Secondly, get your five year old into any English speaking school. You could put your three year old into an independent nursery if you wish.
You are fortunate that both your children are not at critical periods in their education. Some education is better than no education, even if it is not your preferred choice.
It may take time - even a year - but you will eventually get your children placed at the school you wish.