After 10 years in being in this space, I know of one way to track the progress.

If you look at the detail page as a customer, your detail page will have an estimated delivery date for any potential orders that customers may place. Be sure to have an address in the United States as your default address when looking at this so you can have an accurate deliver by date on the detail page. When your item first checks into a fulfillment center, or even before that, you will see that the estimated delivery date for your order (if you were to order your own item) is around 30 days from today. As your items move closer to their final destination fulfillment center, this 30 day turn around time will start to shrink. You can continue checking your detail page on a daily basis and pay attention to this number. Once you see that the estimated deliver by date has shrunk to just 7 days, that means you have around 4-5 business days left for it to be fully checked in.

Depending on where your address is located in the United States and how much inventory you send to Amazon fulfillment centers, you should see a Prime Badge pop up once your items are fully checked in...as well as the deliver by date being 2-3 days from today.