Is there any in universe reason that she couldn't have been training as an MD whilst still nursing ?
Err, because she was busy serving on a starship way out on the frontier at the time? I'm not sure earning an MD is something you could do as a correspondence course.
It's not like you don't have resources on a Starship. You've got a full medical staff, a modern sickbay, a computer with the knowledge of the Federation, a holodeck that can recreate any necessary training scenario including lectures from the galaxy's most renown doctors and an EMH, the earliest version of which was programmed with the knowledge of 47 individual medical officers.
Somehow, I don't think it's like becoming a doctor by mail.