Central London here.
First I have not noticed any food shortages. There may be "out of stock" stickers on some items, such as paracetamol, and a lack of choice of toilet roll (DD is very impressed with the shea butter variant) but we have had no problems with anything else, in any of the probably 50 or more supermarkets within walking distance. I assume that the problem lies with the out of town supermarkets which people drive to where they do big shops.
My two nearest neighbours packing up. Next door has a very very small baby and is running home to mum, which I understand. Her husband has been working away almost entirely since the baby was born and I think she is completely overwhelmed. (I had a mild bug soon after the baby was born so did not visit, and I assume others were reluctant to risk bringing germs into her house.)
The other neighbour is elderly and a widow. I assume she thinks self isolation will be easier elsewhere.
We will stay put. We have a small flat which we used as a base during the decade my mother was living with dementia, which we have not been able to sell. But can't see any great advantage in retreating down there. I think the message is getting through to London faster than elsewhere in the Country.
What concerns me is the idea of being locked in. I work from home, and so am acutely aware of the impact, not going out, even for a couple of days can have. DH and I have been cycling for about an hour each day, as a way of getting sunlight and exercise whilst remaining distant from other people, but guess we will be told to stop. The only thing I think I can do to prepare is to remain as healthy as possible. Perhaps time to dust off my Jillian Michaels DVD (though I suspect it might kill me) and perhaps use any shopping pass to walk 30 minutes to Borough Market rather than the 5 minutes needed to reach the dreary Sainsbury's Local. (No idea whether that will be allowed - or I could cycle.)