This according to this official Swedish website:
http://sweden.se/society/healthcare-in-sweden/
Waiting times for preplanned care, such as cataract or hip-replacement surgery, have long been a cause of dissatisfaction. As a result, Sweden introduced a health care guarantee in 2005.
This means no patient should have to wait more than seven days for an appointment at a community health care center, 90 days for an appointment with a specialist and 90 days for an operation or treatment, once it has been determined what care is needed. If the waiting time is exceeded, patients are offered care elsewhere; the cost, including any travel costs, is then paid by their county council.
Statistics from December 2010 indicate that about nine out of ten patients see a specialist within 90 days and receive treatment or are operated on within a further 90 days [a total wait of 6 months]. Roughly 80 per cent today feel they receive the care they need. In 2006, the figure was 74 per cent.
Also emergencies are seen immediately and urgent care cases are given an appointment date in under a week.
From Wikipedia:
Waiting times
Urgent cases are always prioritized and emergency cases are treated immediately. In urgent cases, the national guarantee of care states that a patient should be able to get an appointment with a primary care physician within 3 days of contacting the clinic.
Sweden also has private medical insurance a person can buy themselves and use just like we do in the US.