In Caretaker Voyager did have an actual medical officer and a nurse assigned to the ship, which with a crew of 150 or so is actually in line for the medical personnel on naval crews of approximate size. Take into account that the ship wasn't even fully staffed yet, was only expected to be out for a few weeks and had an EMH as well as potentially other personnel with field medic training, and they were reasonably prepared medically speaking. Even after they were stuck in the Delta Quadrant, there's an EMH on call 24/7, and Kes and Tom Paris served as his assistants/nurses/corpsman/whatever. In emergencies where sickbay was loaded up we do see other blueshirts doing work, implying there is someone else in the crew qualified to work as an emergency medic.
Similar situation on the NX-01, we do see others pitching in in sickbay during emergencies, though Phlox does seem to be the only medical staff on the ship, IIRC it was stated or at least implied that Crewman Cutler did shifts in sickbay. At the very least she was filling in as medic in the shore leave episode where Phlox was taking his required annual hibernation.