Western medical facilities and hospitals in Tibet are limited. Travel insurance is highly recommended. It is a very good idea to make sure that your travel insurance includes the option of emergency medical evacuation by air.
Please read this summary of hospitals in Tibet at Kotan.org.
Hospitals in Lhasa
TAR People's Hospital
The TAR People's Hospital is a major hospital on the Linkuo Beilu in Lhasa. The Emergency Treatment Centre (Ji-jiu Zhong-xin) is next to the main hospital building. Doctors do speak English at this hospital, ask the reception staff to page an English-speaking doctor.
Chuan-wang Zhen-suo Clinic
Doctors still make house calls in Lhasa, and are very helpful indeed. The Chuan-wang Zhen-suo Clinic is a a few minutes' walk from the Yak Hotel at 100 Beijing Dong Lu. Ask at the Yak Hotel's reception, and they will give you directions in English. Through the reception at your own hotel, you can even arrange for a doctor to visit you (or a friend who's unwell) at your hotel.
Tibetan Traditional Medicine
The Tibetan Traditional Medicine Pharmaceutical Factory was established in 1696 and is Tibet's largest centre for the manufacture, sale, research and development of traditional medicine. The centre produces more than 350 varieties of traditional Tibetan medical formulations. There are many students, as well as a facility for diagnosis of diseases and the sale of appropriate medication for your ailment.
The Tibetan Traditional Medicine Pharmaceutical Factory can be contacted by phone at: 0086-891-6821721 and 0086-891-6813363.
It is an impressive, modern facility. As with most places in Tibet, few health professionals or students speak English. However, the director of this institute speaks excellent English, and is very approachable and knowledgeable.