I cannot speak much of Germany or the United Kingdom, but I've spent a few years in Spain and shadowed there and their system seems to run mostly
well and is well-rated within Europe. It is single-payer and runs through something called Social Security (not the same as American SS, it's like the public health insurance). Expenses per capita are, as everywhere, much lower than America's, and everyone is covered by the public system (through social security taxes paid either by your employer (off your salary), the government (if you're unemployed or make below a certain amount) or yourself (if you're self-employed). The private sector exists and is very attractive to doctors, but where I shadowed doctors seemed to do both public and private work.
Outside of big-name private hospitals, however, it seems the public system does a much better performance (especially got that impression with births,
with some doctors explicitly recommending to pregnant patients that they go through the public and not the private hospital). There is no copay, which has been an ongoing political issue because some people believe it leads to overuse of the system and others that using a copay, even if small, will deter people from using the doc when they actually need it. Wait times are a common complaint, but as someone already mentioned it's not an issue for urgent surgeries. Instead, in the OR where I shadowed, trauma surgeons were operating on old ladies with broken hips from sunset til the wee hours of the morning
Like Canada, Spain has its healthcare broken down into autonomous communities (similar to states), though in recent years policy changes have made it so insurance and patient records are accessible outside of each community.
While systemic issues are galore, as everywhere, there does seem to be some government responsiveness to patient complaints, and the major parties tend to have opposing opinions on where healthcare reform should go (like everywhere, I suppose).
Naturally, doctors don't make American salaries, and it's a common complaint (along with the high doctor supply, and limited jobs, can you imagine that in America?) that drives a few doctors to move to UK or Germany where they're better remunerated.
Although by no means do Spanish hospitals beat their American counterparts in cancer 5-year survivals and a few other statistics, infant and maternal mortality are much lower than in America, and, as goes without saying, 11% of Spanish citizens aren't uninsured. Administrators are also very big on healthcare access to underserved populations, and health clinics are staffed even in the most remote places.