Lots to discuss here. First and most important, your daughter and her partner are the parents and need to sort the childcare. Your original post reads as if there is only one nursery. There must be others. Childminder? Day nanny? Live-in help? Mix'n'match.
You don't say how old you are, but I imagine you are old enough to get tired. And you mention a neurological condition. Ask your daughter and partner if they'd hire someone of your age and with your health problems as a nanny? If not, why not? Ask them if they are prepared to pay you the going rate. If not, why not? Ask them how they'll feel if you drop the baby, get ill, have to go into hospital...
I hope I've read your post correctly. You and your daughter's father are no longer together? I'm wondering if your break-up led you both to indulge your daughter. Your doing three weeks' full-time childcare, and her stepmother (?) flying in from Spain for a week to help you be her nanny, looks like this.
Who brought her up? Why does she expect you to give up your independent life and 'tell you off' if you resist her demands? Why isn't she thinking like a grown-up?
Both you and your daughter need to face up to the reality of the situation. You have, it seems, a month to sort something out. She is walking all over you, and you are letting her. I can’t see 'gently' is going to work. Get out your diary, plan in your morning part-time work and your evening teaching, the Bank Holidays when your daughter and partner will look after the baby, and offer her what you want to of the time that's left. Don’t let her bully you.
If your daughter and partner were killed in a plane crash, I can imagine you would take in your grandson full time; but they haven't been. What would they do if you were killed in a plane crash? They know they'd use their joint incomes to buy in/sort out care for the child they brought into the world.
Be brave. Speak up for what YOU want, don't accept a situation you are not comfortable with.
(With my husband, I do childcare regularly for primary and pre-school grandchildren from two families, and we value the bonds it gives us with them all; but no one has ever stamped their feet when we have said we can't do something.)
Very best wishes.