"I've already sent out invitations, not just to family members but to friends. I can't change up on them now. I would love to have all of you there, but if you can't make it on the day we chose, we will miss you but will make it a friends party rather than friends and family. We'll miss being at Niece's party if you have it that day, but maybe we can get just the kids and us together at a later date and celebrate everyone's birthdays together." Then do just that -- offer specific dates and times LATER for a family meal out. Not a party.
If they fuss and fume, you can add -- if you can manage to say it VERY sweetly: "I know you said you might have your daughter's party that day. I hear you there. But I have already planned and sent out invitations for a party that same day and I am hearing 'might' from you, so we're going to go ahead since things are already set. It really does mean a lot to have invitations already in people's hands and we know people are already setting the date aside for this, so we don't want to move things around again."
I would not worry about the "maybe they'll choose another time that same day and both parties can happen." For real? Think how exhaused and cranky all these kids will be if they do two parties in the same day. The adults will be just as cranky.
Have your party when you want to have it regardless of what they do. By saying "maybe we can do both" you're still caving in to what sounds like a bossy SIL and MIL.
I assume all the kids involved are pretty young. Please believe me, as years go by this just will cease to matter. Once the kids are about four they are much more into seeing their friends at parties than seeing their aunts and uncles and cousins, and the older they get, the less they will care about a family party and the more they'll want to do other things for their birthdays.