That's not how this works. If you are from outside of EU you are not limited to private doctors and hospitals - you can still go and be seen by any NHS doctor, but you have to disclose the fact that you are not covered by national insurance - they will simply ask you for personal details and send you the bill few weeks later(well, if it's worth if - if you're only seeing a doctor about a cold or a broken wrist they will not even bill you anything because it's not worth the time spent filling out documentation, and pretty much no hospital even has a billing department).