I am so sorry. That is the same environment I grew up in, with a mom still reeling from a traumatic childhood who flipped out at the seemingly smallest things. I know how embarrassing it is when other people witness it, and totally get the need to try and shield others from it.
I also know what it's like to be on the other side, as I suffered a great deal of trauma as a child and teen (in addition to the emotional abuse from my mom). I honestly thought I could ignore the horrific things I endured, shoving them out of my mind and moving on without ever acknowledging it, but the visual flashbacks and body sensations came roaring back after the birth of my first child. They got so bad after the birth of my second that I couldn't keep quiet any longer. I realized that I had turned into my mother: snapping or even screaming at people, blowing things out of proportion, jumping out of my skin if surprised or startled and then flipping out because I didn't know how to relieve the internal pressure of the intense terror I felt, etc.
I feel for you, but I also feel for your wife. She is hurting... badly. She needs and deserves help; life shouldn't be like this for any of you. Does she have access to therapy? Would she go? Can you encourage her to find somebody and then help her set it up? If she isn't interested, could you two start couples therapy together?
My mom is old fashioned and thinks therapy is only for "other people" who are obviously defective. I know some people honestly think that and won't give it a chance. I've been in therapy for almost 4 years now, and I'm so grateful for how it's changed my life. And my family's. I'm still processing my trauma, and will be for a while, but my husband has started thanking me for my more positive attitude, increased patience, greater participation in family events and life in general. He tells me that he likes that I'm smiling more these days, and I've got to admit that it feels better for me, too! Life is not perfect by any means. But it's so much better than before. It felt like a big risk at first but it was totally worth going outside of my comfort zone and entering into therapy.
Other, non-therapy ideas include helping her needs to be met. Does she get enough sleep? Does she wake up feeling rested? I had awful insomnia for many years, and it really helped when my husband stepped up and helped me sort it out. Many times I was too overwhelmed to even know what to do. I appreciated when he guarded my sleep for me by sleeping on the couch and using the other bathroom when he had an early meeting that would otherwise wake me earlier than I had to be up, and when he helped me darken the room (better curtains for the windows, taping a note card over the smoke detector light, finding an alarm clock that also projects the time on the ceiling with a much dimmer light and then covering the main display, etc.) It also helps when he takes over bedtime and nighttime duty with the kids so I can try to get a full 6-8 hours of sleep. It's great that he encourages me to take care of myself by giving me time to take long, hot baths (maybe get her some epsom salts and lavender essential oil and help her build time into the day to relax a bit. I use a cup or two of salt and about 4 drops of oil. Add a good book and it's heavenly.)
I know it might be difficult to cater to her when you're ticked off because she just screamed at you for breathing wrong, but that's when she needs empathy and kindness the most. Try to muster up as much understanding as you can, either ask her or assess what might be helpful to lighten her load, and then genuinely try to help her. I can guarantee that she doesn't want to be like this. She is too overwhelmed to do this on her own, as evidenced by her loss of control. I'm guessing she wasn't like this when you married her, and I think it's safe to say that she doesn't have to be like this forever.
Best wishes to you, your wife, and your kids. Please feel free to ask me any questions if there's any way my experience can help you through this. I'm happy to try and help if it means even the possibility of helping to relieve somebody else's pain.
Ps, huge high five to you for loving your wife and reaching out here. I can imagine how frustrating things must be for you, yet you come across as respectful and genuinely concerned about her. She's a lucky gal to have your love and support :)
