Actually she says she works PT because of childcare costs. I would take from that statement that there is still some childcare costs, which are not cheap, even if lowered because they aren't FT costs.
I don't live in London but I do love in a very expensive non SE area, and our childcare is reasonable in cost, comparatively, for the area we're in. For three days a week it's £800 a month.
If we were renting and the cost of that was the same as OP, that's £2,600 on rent and childcare alone. Then you have to add on Council Tax, gas/elec, water, all insurances, car running costs (minus fuel), commuting costs, TV licence (unless we're also counting Freeview as a luxury now...). So, let's be conservative and say that's another 700 per month there, so £3,300 now has gone out already and no one has eaten yet.
Food, let's say a really strict budget for a family of 4, £150 a week including basic toiletries and cleaning things (which is difficult with the cost of living these days). £3,900 gone out now (4 week month). Petrol for days out, general running around, probably around £60 per week as OP said it's an older car between them, so not massively economical. That's another £240. So now we're just shy of £4,150 going out each month. Just over 1k left on your estimates
Kids cost money. They go through shoes and clothes like no one's business. Smaller ones there's nappies to think about. Bum cream. Bubble bath. Calpol. Activities, even free ones, cost something. Colouring in, drawing, painting etc there's materials to factor in. Bubbles, another couple of quid. Ingredients for baking, even cheap ones, another couple.
Then non standard stuff. Birthdays, both immediate and extended family. Collections for people at work. Rainy day savings. Prescriptions/medicines if you're unwell. An appliance breaks down. Car needs servicing.
Yes it's more money than minimum wage. Yes, those of us on it should feel luckier than those on less. But it doesn't go as far as you think it does.