End result would be roughly the same.
In some parts of the country it may be a longer wait than others. The NHS generally though are very good when you visit with very young children.
I've been twice to Danish hospitals, and I think the experience is very comparable. The staff in Copenhagen seemed kinder (a bit more time to reassure the patient etc) but otherwise there wasn't much difference.
I trust both countries in 2015, and Denmark in 2020. I don't know what's happened in Britain since 2015. My family aren't complaining (at least, no more than the usual British amount), but the British people keep voting against their interests.
And that is one of the many reasons why US healthcare is so expensive. I live in Sweden now, and many of the 'minor' things a doctor would handle in US is handled by nurses here. Personally I've never felt this to be a problem at all, but I know many people in the US feel very uncomfortable about that idea.
Coming from Germany and now living in Denmark, I have full confidence in the systems here to handle it equally well (maybe with less than 8 people in a room though).