Not hectoring, but asking basic relevant questions. You didn't give all this info in the OP, so yes you did some drip feeding. About the partner, the abrasive mother, etc etc
In regards to her parents, she either needs to demand that her family step in and help more, or she needs to go find out what local support is available to them. If not, she'll end up carrying on as she is. So rather than badger her about learning to drive, perhaps you should encourage her to locate some local support if she can't get her siblings to help.
You say you've never mentioned anything about her partner, but think back carefully on your reactions and how you discussed things. It fairly DRIPS from your posts that you think he is dreadful. I would imagine some of that comes through in your conversations. Things like that are hard to hide.
So she's your friend and you're there for her, and feel she's having a rough time, so you tell her to get her bum in gear and get driving lessons????????? Okay, let's review. She's working TWO jobs, caring for her elderly parents, caring for her children, and you want her to add driving lessons to the mix?? When is she supposed to have time to do this??!?!?! And if she is working two jobs for their regular household budget, the likelihood of her having the money for the lessons is pretty slim. But hey, no pressure on your friend, right? Just bang on at her for something she doesn't have time or money for.... 
So what if he's sleeping? Do we know what time he gets up in the morning for work? Are we all obliged to be awake until our partners get home from work or wake up when they get up to go to work?!?! Good grief, they're not joined at the hip, you know. They're ADULTS!! She is quite old enough to walk home on her own. She is not a child. My DH works early hours and used to work nights, so was walking for 1.5-2 miles five nights a week, quite late, to work. I drive our family car, but I don't drive him to work. He's an adult, who uses basic safety precautions and is sensible. He now cycles, but again, that was his choice. Does that mean I think my sleep is more important than his? No, of course not. It means I get precious little sleep of my own, and sleep when I can, so that I am also able to function during the day.
We don't know that horribly much about his job, their finances, their relationship, their arrangement for transportation and childcare, etc, to really say that he's being unreasonable IMO. I do think that you are being rather pushy about it though. Ultimately, she and her partner need to make these decisions. Not you.