I'm a green card holder who enlisted Army, but I want an early ship date. How long does it take to get approved for a renegotiated contract?

Don't know. The ability to do the Reno for a new ship date is still a new thing. I've sent 3 requests up before Christmas and haven't seen one back yet. My suspicion is the ability to find a training seat is the hold up. It'll happen sooner or later

Honestly, not long. It really depends on whether or not there's a slot open. So if someone ahead of you drops out or gets hurt etc, then they'll need a body to fill that open slot. Talk to your recruiter and they should be able to assist you if you are trying to ship out sooner.