I've been using a similar place; a Sainsbury's convenience store just down the street. It's small and therefore pretty much impossible to distance staff from customers, but they are being pretty good about only letting a few people in at a time. I tend to be pretty careful in there - mask, hand sanitiser on the way in and out, quick disinfect of tins, bottles and so on when I get them home - but I think I'm probably being overly cautious. If it was a significant transmission hub you'd expect that to affect the staff, but none of the regulars seem to have gone off sick, and although they're taking precautions they don't seem as concerned about the situation as you'd expect if the virus was circulating among them.
In general, though, I don't think British supermarkets have been as assiduous about their precautions as in some other countries. My sister tells me that where she is in Germany you have to use a trolley rather than a basket, and it's taken away and disinfected after every use. The shops I use aren't doing anything like that.