P.M.
At your son's age, verbal correction and redirection may not be enough. Time outs aren't effective for all children, and generally aren't considered to be worth much before 18 months or so. Warnings can easily become just another useful way to get Mama's attention. Tough, huh?
What may work, with a couple of weeks of persistence, is a five-pronged approach.
1. Be on guard – for long periods of time, ready to intercept a hit (this is termed "shadowing" an aggressive child). It's a pain, perhaps but probably less so than losing your job. Stay near enough your son that you can watch him, inconspicuously, out of the corner of your eye. Pretend you aren't watching him at least some of the time, or he'll think he has to behave only when you're on guard.
2. The moment he hits or throws something at the other child, speak sharply and sternly: "NO hitting!" No warnings here; act on every offense the first time. If he gives any warning that it's about to happen, stop him first.
3 (or 4). Hold his offending arm gently but firmly, and direct him on nice touch, soft touch, or gentle touch. It may help to use the same term each time. Use his hand to stroke the other child. Tell the other child "Sorry, Sammy," in a tender voice. (I don't think you should "make" your son say he's sorry before he actually is – it's teaching a convenient lie.) When your son has gently stroked the hurt place, with your help, give positive messages like "Good. That's right. I love it when you are gentle. Oh, that's such a nice touch!" Give your son positive messages so he won't want to settle for scoldings for his attention.
4 (or 3). If the other boy is crying, get between them and comfort him FIRST. Let your son know that when he does this, he ends up waiting for your attention.
5. Be proactive about your son's developmental need to use that over-arm throwing/hitting motion. He's developing brain/body coordination, and that's a very common way for children to work on that at his age. So make up throwing games, with a soft toy and a basket or box as a target. Use long, soft objects to hit an approved target. Wear out that impulse! You can incorporate this during an after-hit session, but it might be even more effective if you introduce it when your son has been playing peacefully so he won't begin to think of it as a reward for hitting.
But try the shadowing – don't be distracted by TV or phone so you can really attend to the behavior. That's what you're being paid to do, so give it everything you've got!