My family is involved with a care home in London which has been let down by a local supermarket which had promised "priority" deliveries for many essentials. The reality is that it is not entirely the fault of said local supermarket - they may have promised to prioritise, but there is simply nothing in that part of the supply chain to fulfil the order.
Two of us spent several hours yesterday searching supermarkets in a 10 mile radius for essentials including bread and milk. No eggs to be found anywhere whatsoever. Not a lot of anything useful, really. It came very very close to the residents going without the bulk of their breakfast today.
As you can imagine, such a facility does not have infinite capacity to store food, even with the best equipment and contingency planning (and there's a ton of it). It's been long enough that supplies have started to run seriously low.
I felt bad lifting the maximum permitted amount of bread from a store near us, chancing upon a batch straight from the oven. It would have looked like I was panic buying, but the reality is that some places are really, deeply desperate.
(There is little evidence of supermarkets pooling resources on the frontline - I can understand it might be OK within factories which supply several chains, perhaps using generic packaging and so on - but the logistics of merging deliveries for Sainsbury's, Tesco, Asda, Morrisons, Waitrose etc. is not to be sniffed at, and probably something they don't actually have the manpower to do at the moment).
I don't understand why anyone would panic-buy eggs and milk. The UK has a large amount of chickens and cattle (you only have to see how many places proudly say how they "buy British", including massive supermarkets). The animals aren't just going to mysteriously stop producing essentials, and in many cases, the farmers are basically able to "do" self-distancing quite happily (and many of the collection drivers could, too). Beyond that, if people weren't buying a shipping container of bog roll every time they go to a shop, the supply chain would more or less cope.