I have heard that Dr Catherine is OK, but I called her once and got an answering machine with a response 24 hours later! I am almost sure that the private Medical Centre in Playa Arena that I recommended is open 24 hours. However I suggest you drop in there when you arrive and get more info from them, including ambulance details. They speak excellent English. There is a public Medical Centre in Alcala (signposted, near the far end) and I believe they have ambulances, possibly the nearest to Los Gigantes. Your EHIC should work there and it may operate 24 hours. Again, worth dropping in and checking so that you are fully prepared. The nearest public Medical Centre to Los Gigantes is in the Vigilia Park area (opposite the Post Office) next to the childrens' playground. Take the street off the roundabout, past the two ferraterias (hardware shops) and keep going. They are not open 24 hours though. The problem with the public services is that they sometimes insist on you taking an interpreter for non-emergency consultations. The emergency number in Tenerife is 112 but there is another number 970 688 908. I'm not sure where I got the latter number and it may connect to an English-speaking operator - perhaps someone else can confirm this.
Hope this helps, but maybe someone else has more practical experience of the subject?