Its annual membership costs five times more than Soho House. Fans include Victoria Beckham. And there's a waiting list, obviously. From music rooms and soft-play zones to celebrity parents, Jessie Hewitson and her four-year-old, Morgan, spend a day at Purple Dragon, the country's most fashionable 'family club.

A group of well-dressed women with expensively blow-dried hair are sipping champagne in a private members' club. So far, so Chelsea, except that it's 10.30am on a Saturday morning and the women are at a child's second birthday party.

The festivities are being held at Purple Dragon, which describes itself as the "world's best family club". It's certainly pricier than your local leisure centre: membership costs between £6,850 and £11,200 ($14,000 and $23,000) a year if you have two children, on top of a joining fee of £1,000 ($2,000). The two-year-old's two-hour party is likely to have cost its parents in excess of £5,000 ($10,000).

Morgan befriends the son of a Hollywood actress. 'Can we go to his house for a playdate?'

For your money, you get access to the poshest soft play in the country. And the most secure – once you pass reception, the doors lock behind you with a swoosh, only to be opened by a keypad operated by the receptionist.

Security, privacy and exclusivity are Purple Dragon's key selling points, because among its tiny clientele wearing perfectly ironed clothes are the children of billionaires, shuttled here from home in bulletproof Lexuses by burly security guards. Their playmates are the offspring of celebrities – Victoria Beckham once enthused about "potato printing at Purple Dragon with Harper" – and the progeny of hedge-fund managers and their whippet-thin wives.

On a visit with my four-year-old son, Morgan, I feel as if I'm on safari, observing the One Percent in its rarefied natural habitat. It is a species you don't usually see, since it is notoriously aloof and prefers not to mix with those with inferior bank balances. This self-imposed segregation starts young: despite fees that are five times the cost of joining Soho House, Purple Dragon claims to have a waiting list.

There is a clear hierarchy here, and the pecking order is brutal. Purple Dragon staff delivering expensive lattes to reclining parents receive only a grunt. A nanny dressed in a nurse's uniform tries to talk to her boss and is barely acknowledged, although the woman lays on the charm with another wealthy mum. Another woman beside me is furiously typing on her laptop – you can work in the main play area while an adult Purple Dragon "play buddy" tries to get your child interested in a range of Montessori-approved toys – then gets up, knocking a cushion to the ground. She steps over it rather than pick it up.

Many of the parents have staff to pick up the cushions, of course. As well as a nanny and possibly a housekeeper per child, there are lifestyle managers, nutritionists, private chefs, personal trainers and education consultants.

I am too intimidated by the cushion-stamping woman to strike up a conversation with her when she returns. Morgan, I notice, is also finding it hard to interact with the children. He is in reception at the local primary school and is a chatty, confident and occasionally scrappy boy. I'm a bit worried he will whack a tycoon's tot and find himself answering to their security detail.

Will he find it easy to make friends with a billionaire's child? It doesn't look too promising. In typical circumstances, he finds a kindred spirit and buddies up with them for the day. Here, with a huge number of adults on hand – play buddies, nannies and parents – there seems to be less interaction between the children. Compared with other play groups or on playdates, there is also less fighting: with such a huge amount of expensive and desirable toys on offer, there is no need to compete because there is no need to share.

The club is divided into separate rooms and areas: the main play area; a restaurant with full wine list; a Soho House-like library; soft play; an art club, where life drawing and sculpting take place; a swimming pool staffed by former competitive swimmers; a music room where Mandarin nursery rhymes are played. This space features a grand piano, electric drum kit and a massive gong.

Morgan loves the soft play best. It's immaculate and there are hardly any children. He can't believe his luck when he gets to the trampoline to find only two kids bouncing around. Quickly sizing up the situation, he realises their two nannies are nearby, keeping a cautious eye on their charges. He steals glances at them, trying to gauge what boisterous behaviour he can get away with.

In normal soft-play centres, some nannies, childminders and parents get involved in their kids' play. The majority, however, prefer to have a coffee, stare at their phones or take the rare chance to follow a thought through to its conclusion.

Not here. The nannies, trailing deferentially after their charges, seem on high alert. If helicopter parenting is when a nervous adult swoops in as soon as a child undergoes a modicum of hardship, then this is private-jet parenting. Nannies are the first responders, parents a distant second. In practice, it means the unfortunate staff have no hope of calming down a spoilt and squalling child, who cries ever louder for their parent's attention.

Purple Dragon was founded in 2008 by Sharai Meyers, who used to work with the designer Roland Mouret. After the birth of her first child, Meyers found the transition from high fashion to toddler time rough, so she hit on taking a Soho House-approach to play. Initially, Purple Dragon opened in Battersea, south London. The Chelsea club is the only one at present, but there are plans to expand to New York.

The crowd is high fashion – lots of Golden Goose trainers and couture outfits – and international, with the odd relatively normal mum trying to coax one child out of the soft play while carrying another in a sling.

When a child is not allowed into the ball pit, his Chanel-dressed mum gives the staff what-for.

I've never been somewhere where the kids are so in charge. Twice I see nannies desperately trying to feed children roaming heedlessly around the play area. Rather than being made to sit at a table, one of the kids being chased sits down to play with a toy train, oblivious to the woman with the spoon waiting patiently for him to take a bite. The child looks as though he might oblige; the nanny hopefully raises the spoon – only for the child to decide at the last minute that the doll's house looks far more fun.

A two-year-old innocently wanders into shot just as the Times photographer is about to take a picture and the club's staff are noticeably reluctant to move the child on. One of them tries to reason with the girl, but she's stubborn. I feel my frustration rising. Why don't they just tell her to move?

It becomes clear later on when the photographer needs to take some shots of the ball pit, preventing a young boy from using it for mere minutes. His Chanel-dressed mum gives the staff what-for.

If the children don't get what they want – and immediately – these terrifying parents are more than ready to make their anger known. It seems to be the one occasion when they are happy to shoulder the burden of being the first responder. And it's not just mums: one dad has a very intense conversation with a play buddy, demanding a Spider-Man outfit for his son. I want to laugh. The fact that there isn't one to be had doesn't calm him down one bit. He clearly sees this as a very poor excuse.

Morgan picks up on the odd power balance at play here quickly. He's twigged that the kids call the shots and he's loving it. At lunch – sitting on his bottom on a chair at a table in the restaurant, I might add – I try to convince him to go for the teriyaki salmon mango and brown rice bowl (when in Rome …). He's having none of it, and proceeds to channel his inner oligarch.

"Yuck!" he exclaims loudly, and instead demands burger and chips. After it arrives, he barks at a passing waitress: "Ketchup!" She gives him a withering look. I'm mortified and make a point of telling him off loudly, so the staff know I'm not like the other parents.

But I fear it's something I could slip into with scary ease if this were my world. I can see myself becoming pampered and meting out micro-aggressions to the staff. I felt it start before lunch, when I couldn't get the table I had my eye on. Worse still, it took ages – five minutes at least – to find another that hadn't been booked in advance. I experienced a flash of annoyance; it seemed like a big deal. Only later do I realise the issue was mine.

By the afternoon, I'm slipping into despair. It's noisy and I'm surrounded by wailing brats playing off their parents and nannies against each other. Despite this, it's the kids I have ended up feeling sorry for – something I hadn't expected. One nanny expressed her concern that the children in her care had a vitamin D deficiency, because they never went outside. Instead, they were shuttled from their penthouse apartment to their daily visits to the Chelsea club, where there is no outside space.

I want to go. Morgan, however, does not. He absolutely loves it and is quite possibly having the best day of his life.

And then we discover the cookery rooms. Thank goodness for the cookery rooms. Here a group of four children sit down to make pizzas and smoothies. I'm so relieved to be in a quiet environment with an adult taking charge that I close my eyes and relish the calmness like a mad woman. It's as though life is back to normal. Then I open my eyes and realise the mother of the two children next to Morgan is an A-list Hollywood actress, and she's sitting diagonally opposite me. It's now I want Morgan to display his best behaviour, but of course this doesn't happen and Morgan promptly bursts into tears. Someone has done the unthinkable and given him a blue place mat when his favourite colour is red.

I whisper without moving my lips that he'll have to put up with blue. This isn't acceptable – he has definitely picked up some bad habits during his time in paradise. "But red is my favourite colour," he yells, lip trembling, fat tears raining down his cheeks, casting reproachful looks at the other kids. "And I hate blue."

The son of the A-lister quietly takes Morgan's blue place mat and swaps it with his own red one. The sun comes out. Morgan gives the boy a long look and a cheeky smile, which is reciprocated. It is the first real engagement of the day. He has found his mate after all.

Afterwards, Morgan wants to find the boy in the play area, but I sense it's not the done thing to stalk a celebrity and her children around a private club they have paid good money to join precisely to escape from people like me.

But later on our way out, by chance, the boys are reunited. The actress's son is trying to put his shoes on while holding an ice cream, so Morgan and I take turns holding the cone. The boy thanks us. He seems a lovely, beautifully brought-up boy and not in the least bit spoilt. Perhaps they're not all monsters, I find myself thinking, as the boy walks off with his sister and famous mum, past the kids'-height aquarium stocked with marine fish and out through the secure doors that close with a swoosh.

"Mummy, can I go to his house for a playdate?" Morgan asks as we leave. I hem and haw. "Can I come back to Purple Dragon?" I play for time, fully aware that we'll never be back, unless he makes it big in the City and brings his own children one day, in which case, count Grandma out. Did you enjoy it, I ask him. "It was amazing. I loved it a million per cent." See? He's thinking like a hedge-fund manager already.

Written by: Jessie Hewitson
© The Times of London