Aren't the UK med schools that are now 4 years long postgrad? If so, then you still need to do a bachelor's degree like in the US, then get into med school. Given this, plus what I wrote above, the length of training really isn't all the different between the UK and the US, all things considered.
Also, the NHS is currently in a downward spiral. Just Google all the junior physician strikes that have been occurring. The strikes occurred primarily because the UK government slashed the compensation for junior physicians -- i.e., the government slashed their total compensation, while raising their base salaries, so it doesn't seem bad publicly, but obviously UK junior doctors are up in arms about it, while at the same time not giving more than token respite in their workload and work hours. Also, Google to see all the junior physicians leaving the NHS and moving overseas to places like Australia and NZ to seek a better future. In fact, the NHS even started campaigns trying to bring back junior physicians who have left, and essentially saying things like "it's not so bad in the NHS," since so many of their junior doctors have left. There's (ironically) even debate about privatising the NHS in order to try to save it.
In other words, it's not all a bed of roses in the healthcare systems of nations like the UK either.