How far were you willing to travel? In my local smaller city a skin cancer specialist was also unavailable for several weeks but I was able to take a two hour trip to the Mayo Clinic and got in there 2 days later with a referral.
In the UK it’s controlled at a a national level and you can’t travel two hours with an overnight stay to fix it like you can in the US.
Separately from specialists, urgent care facilities have a very high density here so immediate things can be addressed quite quickly.
>having healthcare universal wouldn’t necessary mean long wait times.
I have yet to see a system where that’s true. A friend got pneumonia in Canada and had to wait 14 hours in the ER before being diagnosed and given antibiotics.
I don’t think you can get all 3 of cheap, good, and fast. That adage applies to everything else in the world.