This is wrong, you can get same-day appointments for anything time-critical including dental. What takes a long time to schedule is mostly free preventitive care.

There is also the same private healthcare that americans enjoy. You have the option of spending $50 to $100 to go to a private clinic and get same-day admission. In the US services like that cost nearly 10 times the price due to the whole insurance bullshit.

I got free same-day heart check-up and $80 paid same-week heart monitoring check-up.

There are horror stories and footage of the UK NHS, but that's mostly just london and in reality pretty much every city that large suffers from that.