Nope, the emergency department of an NHS hospital is absolutely not reserved for things where people couldn't wait one hour without dying (or people who are in danger of death period).
And since we're talking specifically about the subset of NHS emergency department attendees who are more likely to be dead within 30 days if it takes longer than the NHS' target of four hours for them to receive an inpatient bed, I'm pretty unconvinced that the alternative route of booking a doctor's appointment in the next few days to see if he'll refer in a week or two is the more appropriate one...
I know NHS works differently from the US, but the health care providers in the US that are used by major corps (IE, benefits) all have urgent care lines. I've done this repeatedly (to avoid my wife going to the ER with a sick kid). Typically, while it takes a month to get a routine appointment, the kid's doctor always can see them the next day for something urgent.
If you think about it, this makes a lot of sense because ERs are tuned for low latency, low throughput, high cost for emergencies, and if lots of people who just need urgent care go, they make it hard for the emergencies to be seen with low latency.
And since we're talking specifically about the subset of NHS emergency department attendees who are more likely to be dead within 30 days if it takes longer than the NHS' target of four hours for them to receive an inpatient bed, I'm pretty unconvinced that the alternative route of booking a doctor's appointment in the next few days to see if he'll refer in a week or two is the more appropriate one...