That's more "injury" than "accident". And what about illness? Surely people also go there if they have a high fever, or are unable to stop vomiting, these illnesses may not rise to the level of an emergency, but they're not something you feel like waiting a while for an appointment for. So, you could still classify them as a mild emergency, but they're certainly not an accident. If you want to keep a list of things you might end up in that room for, it would be accident, injury, illness and emergency. I may be forgetting other things though. "Emergency" just seems cleaner because it can cover anything that's urgent enough that you need to stay in the waiting room and have it treated today-ish.

And really, it's not the job of the NHS to push back. It's the job of the voters, the people who use the NHS and don't want to see it destroyed. In Canada we have provincial level care, and they keep trying to starve that too. Fortunately, it isn't working very well. In fact, there has been a very slight expansion of the care the government provides lately in that some dental care is now covered too. But, idiots keep thinking that a partially private system can work. As soon as something becomes partially private, the rich people start using it and then lobby the government to try to avoid having to pay taxes on the thing they're no longer using.