https://en.wikipedia.org/wiki/New_Spitalfields_Market https://en.wikipedia.org/wiki/Billingsgate_Fish_Market https://en.wikipedia.org/wiki/Smithfield%2C_London#Market
You have to arrive at these markets extremely early in the morning. This is the case with any b2b market, e.g. Tsukiji Fish Market in Tokyo:
In the meantime if you're into cocktails the bar manager of Nightjar (same owners as Oriole) has some great videos online: https://www.diffordsguide.com/g/1151/tonys-cocktails
I am not affiliated with Oriole or Nightjar in any way, just a huge fan. (So much a fan that I've flown in to the UK on several occasions just to go to these places, can't praise them enough!)
What I'm saying is that London cannot compete with countries/provinces where plums or what have you grow just a 15-min van drive away from said restaurants, and as such calling a city like London "the food capital of Europe" is a joke. Anyway, also calling Napoli the "food capital of Europe is a joke" because of its size and population density, I can assure you that a 2- or 3-hour drive to the South can get you to some random Calabria small town with way better food than you can get in most of Napoli.
But the food-mania is high with many people who think that just because a place happens to have money it also has good food.
London is only 35 miles away from the sea, how far do you think it is to a field.
As to your Van Drive: For example, Brixham (160 miles) is a popular place to get high quality seafood for the posher end of the restauraunts, this is ~3hr30. Getting fresh ingredients to london is a completely solved problem.