After being approved by OTSG to get a physical, how long does that usually take for my AD recruiter to get this in their work flow?

With the Holiday coming up, your Recruiter will either receive it shortly before or after. All depends on the person(s) sending it.

One thing you have to understand about Army Recruiters is that they have a specific mission they have to meet. These days it's a station mission, but it's still really a recruiter mission. Usually that mission is shipping two people a month per recruiter. That doesn't mean 24 a year. If a recruiter ships five one month and none the next, they failed their mission. So, they do what is called sandbagging. A smart recruiter is going to hold onto a waiver applicant until a month that is slow (like the winter holidays) and process them to ship at that time. So, you'll probably hear from them soon. Just be happy it's not the spring/summer when all the high school grads are shipping or else your recruiter might "be waiting to hear back from ____ about your waiver."

That actually happened to me as a prior service recruit. I waited six months for a waiver that just never seemed to appear, even though I was still in the NG on AD and had just returned from OIF 2. Somehow, all the other applicants had their waivers approved within two to three months, but mine took six..... There is a grading system for recruits and if you are prior service you are a lower value than a recent high school graduate with a 50 ASVAB score, so you should understand that your Recruiter may have a higher priority to put in higher value recruits.