Oddly enough, I'm covered by French SECU and had to use my EHIC card last time I was in the UK. The equivalent of 999 would usually be 18 (firemen) or if in doubt 112 is the EU-wide emergency number. For an ear infection, I would guess any "généraliste" (GP) or "pédiatre" (paediatrician). Bear in mind that in France you pay up front for your care and would subsequently have to claim back from NHS (I imagine). Doctors around here (Paris region) generally cost between 25 and 50 euros. For accident and emergency I was billed about 12 euros, the one time I had to take DD. Couldn't say for appendicitis what they would charge you but certainly any A&E dept. Perhaps someone who holidays in France rather than living here could chip in...?
BriocheDoree - do you know how you go about claiming back from the NHS? We've just moved here and my husband is on an expat contract so does not pay Social Security in France. I haven't been able to find anyone here who can explain the process of getting set up with a family doctor here for someone in my situation. Thanks!