First of all, congratulations on your apostasy. Even if you remain for the rest of your life a closeted atheist it's still better than being a guilt-soaked deluded Muslim
. I'm sure that you will find peace within yourself.
I took interest in your topic because I, like yourself, am a an Arab Muslim-born atheist. I was born in Iraq and lived most of my life there but I also have lived for sometime in Egypt.
My advise to you would be to try as hard as you can to emigrate to a Western country. I'm aware that the process is quite unfeasible, expensive, and can simply take the best out of you. I'm also aware that you might love your country despite its social norms and that you might be too well-established to risk it all. In fact, I'm afraid I'm stepping out of line here. However, I would argue that it's the ideal solution for your dilemma.
But now that you're there, I urge you not to tell your children. Like you said, they're too young to understand. Plus, it's too risky. Being kids, they might very well just inadvertently tell their friends who, in turn, might tell their parents. Then, only terrible consequences would ensue. Not only your children will feel different they will simply be social outcasts. The self-righteous parents will accuse you of satanism, decadence, and polluting children's minds.
At the same time, there are lots of things you can do to "ease them into" atheism. Tolerance, open-mindedness, and religious moderation should be cornerstones in their upbringing. Try to get them interested in music maybe. Avoid religious friends and families. By doing so you can at least make sure they don't grow up to be religious nutters !!
Hope everything goes well.