Just a data point:

Submitted paperwork on 6/3. I used our tax returns as a way to show we were all family (myself, my daughter, and my wife). I blacked out the last 7 of our socials per the website instructions. I also submitted the family plan application.

On 6/4 I got a an email saying they needed photo id. It was a standard email where they put an "x" next to the part that you were missing.

On 6/5 I emailed in photo copies of our passports as well as our family plan application and tax form that was already submitted to [email protected].

On 6/6 got an email saying everything was received and my request was approved.

Today (6/7) I logged in and see my wife and daughter listed in the family plan section.