First, take a deep breath. You are so stressed out and that is not going to help your health.
Then remember that your husband stepped up and stepped in -- he took on SIL and now she, not you, is dealing with things. That was his role all along, and he frankly should have done it long ago, but he's done it now. TELL him that you recognize that and tell him what it means. Also be clear with him that the situation must stay that way; SIL demanded that you butt out; he told SIL to deal; that needs to stay in place. You need to focus on your own health so you are around for your own children.
Refuse to take MIL anywhere (short of a call to get her to the ER -- if she's 45 minutes away, call an ambulance if she calls you to go to the ER!!). I would guess that at some point, SIL or MIL is going to call you and say that car service is overbooked, or something came up, or SIL forgot to book a car or make an appointment, etc., etc. It is tough -- because MIL is indeed seriously ill -- but you need to step back when that call comes and say, "I have some health issues that must be dealt with. I'll have Husband call you." And then let HIM handle HIS family from now on.
I would help in one way, though: Gather the names and numbers for several different car and taxi services, etc. and have them ready where you and husband both can e-mail them instantly to SIL or give them to her over the phone. That way she can't say "...but I don't know what to do."
The other alternative would be for your husband (not you! not you!) to look into giving his sister an old, used car that works reliably enough to get MIL to appointments around town. Do not expect this to be a matter of "you can repay us"! Make it a gift or don't do it at all. Adding a money debt to this mess would only make things worse.
Regarding the uncle, if he is "kind of backwards" (and I am not sure what you mean there -- health issues? Mental capacity?), if he needs care, someone should be looking into a facility where he can be cared for, period. If he's MIL's brother, sick MIL has no business trying to care for him, obviously. He might need either in-home care or might need to find a place to live where he can be cared for if he is not capable of self-care. Seriously, someone professional needs to evaluate him and start exploring the options for Medicaid, or assisted living, or home aides who come into his home -- whatever he needs, but it's impossible to tell from the post; however, you should not be doing all this. Husband needs to get involved here if uncle has no kids of his own or others to do this, but it will be a lot of work up front and uncle (and MIL) might balk hard at the idea of intervention.
One other thing. SIL's blow-up at you about butting out, etc. was clearly so painful and wrong. And you and she also clearly have years of issues and resentments between you, especially as you feel she has not carried her weight caring for her mother and uncle. This is all why your husband now needs to intervene but not by doing all the care and transportation himself--but by making arrangements and getting information that would lead to the best care for MIL and uncle. MIL sounds as if she might even need assisted living that has a medical component. But it's husband's and SIL's role to do that -- or if not, SIL should accept that she is now a full-time caregiver.