The problem with a "tough love" approach in this situation is that the innocent ones are the ones that would pay the most for it (the children). While it's true that there are many wonderful foster homes out there, there are also some scary ones. Also, once those kids are in the system, it could be challenging to get them back out. Can you imagine the feelings of desertion those kids would feel? If any of my cousins' or sibling's children needed care for any reason (even if I was being "taken advantage of") they would have a home with me. There is no way I would put them in the care of people I didn't know. Your mom is to be commended for what she is doing for those children and the she compassion obviously has for them. That being said, she is obviously in a bind financially (and emotionally) caring for them. If it were longer term, your mom could possibly be appointed as a foster parent and get some state aid in caring for them, but that this is a short term problem probably takes that option away. It could be an avenue to consider if the situation continues, though. The parents need to be honest with themselves and everyone else about why it's taking so long to get out of the house. If it's dangerous to work in that could account for a lot of it, but if there is some other reason they need to figure that out. Perhaps she needs a couple of days with a ton of help to get the move over with (like an old fashioned barn raising). Also, if the husband is controlling, she may be getting overloaded with crap from him to take care of. I agree with you that in a perfect world, they would be caring for their kids more at the same time as moving, but there is obviously more going on.
My recommendation would be to work with the wife and set a date (very soon) and gather all healthy bodies available to get a move over and done with. The husband may just need to get over the fact that he may have to live in something less than "perfect" (a mobile home) and the wife may have to be prepared to have him lecture her about it. But, it's time she pull on her big girl pants and take care of her family's needs, not just her hubby's wishes.