so, in this situation obviously it sounds like your mom can't be babysitting anymore, and you've already decided that she won't be. that doesn't mean she can't be a loving grandma with supervision though.
we have self-closing gates, and I've gone upstairs to the bathroom and come out to find DD had come through the gate at the bottom of the stairs multiple times. I've forgotten to close the gate that leads to the basement multiple times. and DD has fallen down the stairs when I was right there several times. she's also tumbled off the bed and off the couch. when my MIL was visiting they were sitting on the balcony chairs and DD fell off and scrapped her head really bad. two weeks ago she fell backwards off my lap and hit her head on the coffee table. no one was being neglectful, she was never seriously hurt, accidents happen. saying that because your mom was in charge when your son got hurt she can no longer see him is just outrageous, and I would tell your DP that kids get hurt, it wasn't malicious, and you see no reason your mom shouldn't be able to see your son as long as someone else is present and responsible for safety. it also sounds like there's bad-blood between the two of them, and that needs to be sorted out. whether they like it or not they are family now, and they need to learn to deal with each-other in a civil manner (I'm not saying they need to be best friends, but they need to get to the point where they can be in the same room and act civil towards each-other).