Unfortunately for you, you cannot dictate who he has in the house when it is his custodial time. If you are going to his/the kids' home and she is there, then she should be the adult in charge and you stay home. It sounds like you and your ex had an agreement that only worked when no one was dating (which is my big issue with the idea of nesting which is what it sounds like here).
Just because you can't imagine it doesn't mean it's wrong. Would you be this upset if she was a live-in nanny or au pair? Are you just peeved because she's his GF or are you peeved because she's only been his GF for a few months? I wasn't in the same situation, exactly, but as the GF and now wife of a custodial father, I was OFTEN in charge of "someone else's kids" and many times it was just because it was the right thing to do to help him or them out. No need for the kid to sit at daycare when I was capable of getting her earlier and start her on her homework.
If this routine no longer works for you/him, then you should make a more conventional custody schedule and then each worry about your own time. And completely separate houses.
I also caution you about being overly worked up about what a grown man does. There is only so much you can legally control (we would have liked it if DH's ex actually divorced her current husband and didn't use her boyfriend to transport her teen daughter when she barely knew him) so stick with the things you can actually do something about and make sure the kids are able to call you if school time arrives and they are left without a way to get there. If they cannot get to school without you, then why not go to mediation and ask your ex for a more traditional every other weekend routine? He might need to change his job schedule if he cannot get the kids to school on his own time. One reason my DH's ex had weekends and summers and breaks was because she was terrible at anything routine, so she wouldn't get them to school on time (if at all). She only had time when getting up at 6AM didn't matter.
Look at it as finding a good schedule vs "taking the kids away". Please avoid making your kids pawns. That will only backfire and hurt them more than your ex.
ETA: When my DH was single, he hired an au pair. His ex had nothing to do with it. It was his decision for his home for his children on his custodial time. Sounds to me like Mom is mad because she's being replaced and she needs to re-evaluate the custody situation and this business of going to the former home every day. I think the parents need to make a formal and well-defined split. THAT is the real problem - that things are not clearly defined.