I would say nursery duties have to relate directly to the children. My list would be:
Care of children's clothing, including washing, ironing and putting away. Polishing shoes. Organising wardrobes. Buying new clothes/school uniforms. Basic repairs and sewing in labels.
Care of children's rooms, including hoovering/sweeping playroom and bedroom floors. Organising and cleaning toys and books, and basic repairs (sewing up holes in teddy bears, sellotaping torn book pages). Keeping rooms tidy. Changing and washing children's bed linen.
Cooking healthy meals from scratch. Could include cooking extra and freezing. Making up bottles of milk, and sterilising bottles. Cleaning and tidying kitchen after cooking/meals.
I also think buying presents for birthday parties, or odd items like dummies, milk and bread for the day, is reasonable. It's reasonable to expect the nanny to be in charge of administration for the child too (doctor/dentist appointments, writing thank you letters, ensuring permission slips for school trips are signed).
Things like parents laundry, changing parents beds, cleaning/tidy rooms not used by children, full family shopping, organising/managing tradesmen or other staff, cleaning windows, gardening, pet care or dog walking, cooking for parents, running errands for parents, heavy cleaning are not "nursery duties" but some nannies may be happy to do some tasks, or could be employed as nanny/housekeepers.