I understand why you're resentful and I do get it. That comment of your sister's about care of your mum falling to you is beyond infuriating. I can see why you feel she would take advantage if she could. However.
Your mum does not have to take on three under fives in order to make sure that you get a fair deal. It's just asking too much of a parent. Sadly, there is a sense in which grandparents helping with childcare does indeed come on a a first come first served basis because what's the alternative? That your mum didn't help your sisters for all those years because she wasn't sure she'd be able to look after your children as well by the time you got around to having any? That would be dog-in-the-mangerish of you.
It's unreasonable to expect someone to go to impossible lengths just to make sure you get a scrupulously fair deal in life. Your sister has had a certain number of extra years being mothered by your mum (because she was born first) but you wouldn't expect your mum to make up for that in any way. It would all get a bit absurd and your mum is left caught in the middle somewhere, wondering (I'm sure) why everyone wants a piece of her. She doesn't have to do anything for anyone, you know.
This would all be different if you and your sister were having children at exactly the same time and she was choosing one of you. But she's choosing to continue looking after children she views as in need of a mother figure to come home to (much harder to send older children to aftercare/childminder and expect homework etc. to get done) and she isn't doing bottles, nappies, all-day care, buggies, running around, mess etc. It's much easier. In expecting her to drop one arrangement and start another, you're effectively asking her to at least triple her workload. Why should she, is what I'm wondering. So you can not have to spend money you can perfectly well afford to spend? But hasn't she done her bit for you already in bringing you up?! I have two small children and no mother, but I would never have expected her to shoulder any of the burden of looking after my children. I'm grateful she raised me, not looking for even more.
You seem to feel resentful that your mum isn't in a position where she could drop everything to pick your children up from nursery, because she has your sisters' children. If this is really all you want from her, it's not nearly enough to justify turning your niece(s) and nephew(s) out of their stable childcare arrangements anyway. When my mum was alive, I would of course have turned to her in a crisis, with many apologies. But I would never have planned a situation where I'd regularly be having crises and be reliant upon her to drop everything, just because my busy life couldn't accommodate all my priorities. If your mum wasn't doing childcare at 6pm on a weeknight, I would like to think she would be sitting down to dinner in front of the telly, or perhaps getting ready to go to a nightclass, or whatever took her fancy. Because she can. She's earned the right to have no responsibilities for a while.
You haven't said if your sister had as many children as close together as you have done, or how much younger was mum was back then, but in my view, both you and your sister are a bit unreasonable.
When you say that your sister should make alternative arrangements to give your mum a rest or some free time, it seems difficult to reconcile with your also feeling that your mum should be using any extra time she has to look after your children.