This thread has got a bit nasty hasn't it? Don't listen to the extreme views OP - you get them on all forums. There are plenty of balanced ones on here with actual advice that are worth reading.
I would agree that you can't expect any childcare from GP's, but GP's who have zero interest in yours or your kids life should not expect anything from you either. You have kids because you want to be a parent, not because you want a carer in old age. That is cheeky fuckery too.
It could be your son's behaviour that is putting her off. If that is the case, it wouldn't be a good arrangement anyway. GP care, while much appreciated, is unreliable from experience. My mother isn't very confident, but my sons adore her, mainly because she makes an effort, whereas my OH's parents could be cardboard cut outs to my kids as they don't really interact with them much. You are better off using paid for care if she guilt trips you and is reluctant if you can. I don't think your kids will enjoy it much either by the sounds of it.
I would say if you want to move, do it, but not because of childcare. If you don't pay for it, it is unreliable and it definitely has to be a mutually enjoyable arrangement for all, including your kids. The most important thing to consider is which parents are more keen to be a part of your lives as a family if you want your kids to feel more connected to them. If your mum lives near by and barely sees you, it says a lot. Your children will benefit the most from having GP's who are a part of their lives and parents who are less stressed. Does you OH want to be near his parents more? Is that causing resentment? It sounds like your mum is the sort of parent who might enjoy her jollies down to Cornwall for a holiday every few months, so it might be the best for all of you.
So yes, you shouldn't expect your mum to provide childcare, but she shouldn't expect you to stay if you aren't happy and she can't be bothered to spend time with you either. If your family are struggling and you are really stressed due to you both working, the fact that she doesn't even want to come round and help (even if it isn't to look after the kids) says a lot really.