I'd say it's real - I can't see Billie getting her a fake and also Nellie's wardrobe is very expensive.I don’t think it’s Van Cleef & Arpels. Doubt they’d buy her a £3k necklace. They’re stupid but surely not that stupid.
I personally wouldn't trust a 6 year old with necklace of that value but have no problem with them spending it on her as I feel it is all relative to the amount of money you have / can afford to spend.
My son is 6 and has had several pairs of Ray Bans since he was 2, has had Balenciaga Sock Trainers, Gucci Trainers, Moncler Jacket, gucci jumpers etc - stuff I like and can afford without getting myself into any financial stress.
My husband and I have very good jobs - we travel (pre-covid) 5/6 times a year on Disney Cruises, to the Caribbean and Bahamas and to Dubai frequently.
My husbands side of the family wouldn't have the same income we do, prob not even half and have been vocal about how they don't agree with a child having such expensive things when they could be damaged, broken or lost.
Firstly we teach him to look after his stuff and we still have all the Ray Bans, some that are a bit scuffed we use for water parks etc. and secondly I don't think anyone has the right to judge what people spend on items or travel once the people are paying their bills and not getting into debt over it. We also teach my son not to brag and to be grateful for what he has.