I’m not exactly sure what you are trying to describe. Has the first hen weaned her chicks, gone back to laying, and gone broody again? Is she still taking care of her first chicks? I don’t know what your facilities look like either. Are your facilities so small the broody hen keeps her chicks in the coop during a good part of the day or does she have enough room to keep them outside practically all day? My broody hens usually take the chicks out of the coop in the morning and never go back in until bedtime, but I have a large coop and a lot of room outside. I also have food and water outside as well as inside. Your situation sounds unusual to me but we keep them in so many different situations than about anything is possible.
It’s possible that one broody hen will attack and kill another hen’s chicks. It doesn’t happen that often but it does happen. What is more likely is that the broody hens will fight over the chicks. They will both want to raise them. The winner gets to raise them. What is most likely is that the two broody hens will take care of their own chicks and not bother the others. But a lot of this depends on how much room you have. Chicken behaviors are almost always dependent on how much room they have.
I’ve also had the situation where two broody hens fought over the eggs. The chicks inside the eggs of the one that had been setting longest started to internal pip. They chirp when they internal pip. The other hen heard them and decided she wanted those eggs, not hers. They destroyed half the eggs in their fight.
I’m guessing that for whatever reason your first broody is keeping her chicks in the coop during the day while the other hen is on her eggs. The first broody doesn’t like something about this so she chases the second broody out. Maybe the second broody was showing interest in her chicks? I normally don’t push isolating a broody hen when she is incubating but you have to be flexible. This sounds like a great time to build a pen around the second broody so she can’t leave her area and the first broody can’t get in. Use wire small enough the chicks cannot get through. Make it big enough to put food and water in there and keep them isolated in there for a week or so after she hatches.
Good luck!