I don't think you can use other pronouns. It's a fairly fixed phrase. In the Christian religion at least, the belief is that things only happen if God wills it (ie if God wants it to happen and makes it happen). The theory is that no matter what us human beings do or what we want to happen, it won't happen unless God wills it. So in your "driver" example, it really doesn't matter how willing or enthusiastic the driver is, he won't catch up with them unless God wants him to.
I should point out that this is my understanding of it but I'm an atheist so I might not have grasped the nuances of the omnipotence theory (if "omnipotence" is even the appropriate word).
Before posting this, I mentioned it to my mum. She has always interpreted "God willing" as "If God is willing to allow it to happen", so she thought it meant that human beings could have an effect on outcomes although only if God allows their desired outcome to happen!
I considered deleting my reply but I still think the first part is valid - you can't say "The driver willing ..."
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