My daughter has had a similar problem with her MIL. My granddaughter is 4 and myself and her Grandma (MIL) have shared some childcare.
I eat healthy avoid any Ultra Processed food, never have biscuits, cake sweets, chocolate, ice cream, sugary drinks in the house. So at my house GD eats healthy no snack cupboard and healthy meals. I make her Greek yogurt and fruit lollies or sugar free banana and oat muffins or hummus and carrot sticks for a treat. She never asks for anything different in my house.
At her Grandmas she can help herself to crisps, biscuits, chocolate bars etc, my daughter has fell out with her a number of times but Grandma says it’s just a treat and has threatened to not look after her.
Saying that my granddaughter started school on Monday and for 3 years has free school dinners, I have been horrified at the offerings this week, pizzas, fish and chips, all day breakfast and for dessert doughnuts, choc chip cookies - they can choose fruit which she has some days but given the choice most children will pick the sweet things.