Suzi Leather hesitates for a second. Is she worried about the school meals her three children have eaten? “Mmm. My kids have been through state and independent schools. Like many parents we did have concerns about meals in the state sector.” How did she resolve her anxieties? “They had a mix of school dinners and packed lunches.”

Leather is not just any parent, of course. She is the outgoing head of the School Food Trust, set up by ministers to oversee school meals in the face of national horror at the revelations of chef-cum-public-hero Jamie Oliver.

Last spring Oliver exposed in a four-part television series the junk being dished up by dinner ladies. In Greenwich, south London, children were being served lunches whose ingredients cost less than 37p a head. Turkey Twizzlers, chips, burgers, fizzy drinks were all on offer. If some children were bad-tempered, listless, overweight . . . who could be surprised?

“We started from the point that school food is shit,” said Oliver, after struggling to introduce Greenwich’s teenagers to freshly cooked dishes such as balsamic beef and bean wrap. His hard-hitting campaign embarrassed the government but the steps it took last May seemed a bit of a mess and a muddle: an extra £220m to improve lunches, “tougher” food standards and a couple of quangos, including the School Food Trust.

Leather is quitting the trust at the end of this month but not before saying that she wants the government to spend a lot more on lunch ingredients than the “half a quid a kid” that Oliver pushed for.

“I think improving the standards of school meals is going to be a very long process,” she adds. “This is not an easy task. Research we commissioned suggests it will be difficult to deliver the new food standards unless we spend 70p for ingredients in primary and 80p in secondary schools.” In other words: £450m for healthy eating.

Last week Leather released figures showing just how dire things are. So many councils have farmed meals out to private caterers that one in five primary schools no longer have a kitchen to cook fresh food. The number of children eating hot lunches is plummeting, too, as worried parents opt out, some reluctant to pay the rising price of canteen fodder. More than half England’s children have abandoned school dinners, which now cost an average £1.60.

It is pretty bad news for the trust, which has a target of increasing the take-up of school dinners by 10% over the next two years. How will they do that in the face of this mass exodus? The answer, it seems, is to persuade parents to ditch sandwiches. “Research shows the average packed lunch parents prepare is nutritionally poorer than even the current standard of school meals,” says Leather controversially.

One of her last moves at the trust has been to set loose yet more researchers, this time to discover whether a good hot midday meal really does improve children’s behaviour and exam results, as some claim.

As Michael Nelson, the trust’s chief researcher, puts it: “We need to get across that if you pay a little more for school food, it means your child may do better academically and get a better job.”

Parents who do persist with the Tupperware shouldn’t be surprised if they are quizzed on its contents. In Greenwich, where Oliver’s eating revolution has finally taken hold, children tell tales to teacher about suspect snacks.

David Ashley, headmaster of Greenslade primary, says that pupils who bring in packed lunches “are allowed chocolate on a biscuit but not a Mars bar”. If such sweeties are spotted, parents are called in for a quiet word.

At Charlton Manor primary, the head, Tim Baker, says: “Children get stickers for healthy boxes . . . If a child brings in a chocolate bar, we take it out of the lunchbox and give it back to the parent at the end of the day.” Pupils give each other away, he confides: “They say, ‘Miss, he’s got sweets in his box’.”

Does Leather approve of what smacks rather of the nanny state? Absolutely. “If you are producing high-quality food and at the same time turning a blind eye to unhealthy packed lunches there is an inconsistency there,” she says.

Meanwhile, the heat is rising in the kitchen. Oliver is putting the finishing touches to his follow-up documentary to be screened in September, just as the government’s tougher food standards kick in. It is the trust’s job to check on whether dinner ladies deliver these — fresh fruit and veg with every meal and fried food, like chips, no more than twice a week — but Nelson confesses that a new way for the school’s inspectorate to monitor the situation is still being developed.

I try to remind Leather that she once said: “Standards for school food should be the best we can do, not the most we can get away with.” How far do we have to go?

But the interview time is up. My two will get wholemeal sandwiches and an apple for a while yet. And the youngest will still get the odd bag of crisps.