Well we do see multiple people assisting in a few episodes, so I assume that while we don't have a nurse speaking role, that there were people with multiple training fields, who would assist Phlox.
Based on the visual evidence, no staff present during the majority of scenes where we see one person getting treated, and seeing some form of support staff when a major ship crisis occurs, means no full time nursing staff.
So the only real question is, was the personal there in major emergencies part time field medics, or did some have nurse training (or does Starfleet teach some sort of combined role)
Another point is it really necessary to have a full time Nurse (or Nursing staff) on board a vessel with only 80 in crew?
Where my father worked (he was isolated from society by hundreds of miles) and worked with about 100 men and women, and they had one full time medical staff (they were classified as a physician's assistant). Of course the one difference is that they were only a an hour or two away from Fairbanks or 2 and a half hours away from larger medical staff in Anchorage by plane). SO they didn't need a well versed and trained doctor. On Enterprise you would certainly need a doctor of considerable skill and wide based knowledge.
But would you need a full time nurse, or even a full time medic? I would guess no (and its truly a guess) but I would assume they had cross training with a few crew, who's full time job was something else.