At the moment, the school meal service is subsidised by about £140 million a year, quite separate to the similar sum which pays for FSM. This is because the catering provision tends to need around 40-50% takeup to break even, and the average is about 38-39% (more in primary schools, less in secondaries).
So if they use one year's worth of subsidy to improve many kitchens in places which have been downgraded or neglected for years, and then get all primary uptake up to around say 70% by providing KS1 meals free, the amount they have to subsidise the service should go down thanks to economies of scale (it costs hardly any more to feed 70% of a school compared to 40%, because kitchen staff costs and energy are the main costs), while the amount the caterers have to spend on ingredients should go up.
It's been done in various boroughs and schools already, and reportedly been very successful - admin time is freed up because no-one has to chase dinner money or entitledment to FSM, so there's a saving there, and practically all kids want to do what the majority are doing. I know some schools funded it by going vegetarian, which was accepted after they did taster days for parents. There certainly should be vegetarian provision and provision for medically-required diets - don't know about vegans.
Problems like lack of time to eat, kitchens runnng out of best options before the last classes get there, food that has been deemed nutritious but isn't tasty, have been recognised and resulted in the School Food Plan which was recently published, led by the founders of the LEON chain. It's all about improving the eating experience rather than the actual food.
Hope that puts it into a bit more context. I think it's a great idea - but then so were all the initiatives to improve hospital food over the last 20 years, which didn't get that far.