I'm the mum of an older teenager so we're not in this position yet. I have no idea if I'll be ready, willing or able to help with my GC if and when the time comes. I hope I will but, as an older parent, it may be more limited. I don't know.
However, if my child 'threatened' me with 'help me with the grandchildren or I don't help you when you need it' having loved HIM/nurtured HIM/supported HIM and paid for everything he needed or wanted during HIS childhood my reply would be 'then I'll be using the £500K house inheritance you THINK you're getting to pay for my now fantastic care sunshine'. I suggest kids who hold their parents to ransom over caring for the grandchildren remember that when they're parents are elderly....Threats work both ways too.
But then we've brought our son up to appreciate that loving their family doesn't come with conditions or pay back. Hopefully he'll appreciate whatever we can do rather than insisting, calculating or assuming we'll do more.
OP, I wouldn't be spending every Saturday at the GPS as your kids need more variety and, maybe, your parents need a bit of a break from them. Seeing them less often may make it more of a treat when they do meet up, on both parts.
And get a baby sitter if you want an evening out. You can't force them to enjoy your kids, your mum is ill (and may be worried understandably), what's going on with your dad seems very complicated and they weren't there for you as a kid so what do you think has changed?
Forget this stuff about involved grandparents - there's always a study somewhere that says things should be done a certain way. We're older parents as I said, so DS's grandparents have all died. We had no-one to baby sit him except friends occasionally (which we were grateful for). He's pretty well adjusted for not having the GPS input...
And don't compare your mum's ability to go shopping etc with looking after your children. I can easily go out for a whole day shopping or to the theatre because I can pace myself. Looking after a toddler again for the whole day would not as easy even now.