Let me tell you how this worked when my grandfather "loaned" my father money to start a business. It created animosity in our family for years. My father couldn't pay it back. My grandfather said to just consider it your inheritance when I'm gone. My father's brother resented what had happened for the rest of his life which was probably another 30 years. Even tho he received an inheritance equal to or more than the amount my father "borrowed" he did not think it was fair. The two families had hard feelings for 30 or so years. Fortunately the children have not carried the anger after the deaths of the brothers.
Do not loan your sister money! I'm guessing your mother made you her money manager because she felt she could trust you to use her money wisely. Loaning money to a friend or relative is rarely a good idea. It is never a good idea to loan money to someone whose credit rating is so low they can't borrow money any place else.
Yes, your sister is in a difficult place. You can sympathize with her. In reality she doesn't actually need your mother's money. They can sell their house as is. Yes, they may not make any money. They may not be able to sell the house for the amount they still owe. But they can sell the house. If they borrow from Mother they'll owe her money. If they sell house as is they'll owe the balance on the mortage. Either way they owe money. As I see it the only advantage to borrowing from Mother is that they won't be forced to pay her back if this new life doesn't work as they want it too.
And I understand that divorces have the potential of being easier if there are no or only a few debts. They can be divorced. The court will divide up the debts between the two of them.
Your sister needs money to move out with her boys. That is her responsibility, too. Is she working? If she's doesn't have job skills, is she working on getting them? Is the boys' father working? Won't he be paying child support? She is asking for money now. Does she have a plan for making a living once she's out?
My mother as were the mothers of other friends was worried that she would outlive their money. She fretted about it for 25 years or so. I would guess that your mother would also be worried if she knew that she wouldn't have that money. She and/or her husband saved and invested so that she would have money for her last years. An inheritance is just that. It's available only after the person dies.
I'm guessing that there may be hard feelings on your sister's part if you don't give her the money. How she feels is her responsibility. Your responsibility is to preserve your mother's assets for as long as she is alive.
One way to put the responsibility back on them is to ask them how they plan to pay the money back? They are not financially solvent now and have not been for many years. "Future behavior is best predicted by past behavior." A quote from the world of psychiatrists. How can they prove to you that they will be able to pay your mother back? Since their credit rating is in ruins what can they show you to back up their ability to repay the loan?
Money is not a family matter. It's a business matter. You would be remiss if you didn't protect your mother's nest egg.