What are the repercussions to missing AT?

You can expect a phone call at the start of AT, then after AT a certified letter from your command called a Notification of Separation (NOS).
It will detail the entire procedure starting with when your SGLI will end to what your recommended characterization of discharge will be and how that will impact your VA benefit eligibility.
If won't matter if you didn't sign for the certified letter, they will still process the discharge.

You can't RST active duty orders, they have to be amended or revoked. For some units, AT is the only chance they have to complete weapons qualification, among other trainings. Missing it makes you unqualified for the whole year.
Have you requested a partial AT? You'll need to be upfront about your family reasons with your leadership and discuss what portion of AT you would be required to attend.
Failing to attend AT, when you have an approve AT order, is called missing movement and is punishable by discharge.

You’ll be considered an unsatisfactory participant and the commander could initiate separation from service.

Unsat performance. Unsat attendance. Missing movement. Failure to follow orders. A whole slew of issues you don't want to face. Which could all lead to you getting the boot with a discharge type you don't want.

1. "family reasons" is very vague. is someone sick? financial hardship? or did someone just plan a vacation during AT? 2. 36 years and I've never seen anyone get denied RST who had a valid reason. Bottom line: unless someone is sick or dying, you need to go to AT or accept the fact that you will be chaptered out with a less than desirable discharge.

You are risking a lot if you follow through with being a no show.

And regardless if you have a previous discharge if you get booted with an OTH or BCD that will be what sticks

You’re probably going to get discharged. I’ll be blunt. You asked for time off and RST and it was denied. That was your answer. Deciding to not attend after your commander already told you what was expected of you is now disobeying orders.

You need to go to the AT or probably be prepared to be out of the military.

Curious, did you attend the AT?

A real good chance for finding yourself a total civilian with a Other Than Honorable Discharge.