That has nothing to do with Universal Healthcare, and has everything to do with the fact that there are not enough doctors. Which is unrelated to Universal Healthcare -- just look at a country like Austria, which has very good social insurance, and absolutely tons of doctors.
Here, the education to become a doctor is free, and there are lots of (free) childcare services available to average people. So a person need only have enough money to survive, and need not pay for school or babysitting above the age of about 3, and graduate a medical doctor without debt.
That is how my mother-in-law, who grew up rather poor, became a neurologist after she had her unplanned baby (my husband).
Beyond that we (North American) have rules betters suited for an abundance of medial personnel, rather than a shortage: You need a regular doctor visit to renew benign drugs like Ventolin or Insulin, we have nurses cleaning up vomit instead of janitors, and licensed dental hygienists are not allowed to clean teeth unless they have a dentist on site [1]. Which would be fine, if we had doctor, nurse and dentists to spare, but they seem like comparatively easy things to fix if there's a shortage.
Here, the education to become a doctor is free, and there are lots of (free) childcare services available to average people. So a person need only have enough money to survive, and need not pay for school or babysitting above the age of about 3, and graduate a medical doctor without debt.
That is how my mother-in-law, who grew up rather poor, became a neurologist after she had her unplanned baby (my husband).