You are right to worry. If they were your cats you could keep them well wormed, but it sounds as if your neighbours are not the sort to make sure their cats are not a danger to others, including your unborn child, from toxoplasmosis. Of course the contaminated hay is also a danger to your animals, but the main concern here is you and your baby.
Is it possible that your husband could catch the cats and worm them? That would be the first step. He, or someone else, needs to help with decontaminating the barn so you are not exposed to cat faeces.
Once you are down to uncontaminated hay you can cover the stack with a tarpaulin - if it was going to sweat it would have done it by now.
Next you need to deter the cats from thinking your barn is an ideal place to hang out and to use as a toilet. I am a great believer in the water pistol - it doesn't hurt the cat but they hate being splashed, but it does mean you need to lurk around the barn for a few days armed with the water pistol.
Legally, I think cats are exempt from a duty of control so your neighbours' attitude, whilst unpleasant and irresponsible, is within the law. However, I'm not a lawyer, so it could be worth seeing if you could at least compel them to keep the cats regularly wormed.
For your peace of mind, it might be worth taking a sample of the cat faeces to a vet to see if they are carriers of toxoplasmosis, and if they do have worms - this would be fuel to use in your discussions with your neighbours.