LONDON—Amazon.com Inc.’s voice assistant Alexa will provide users in Britain with medical information from the country’s National Health Service, in a deal that broadens the tech giant’s push into health care but has privacy advocates worried about the possible misuse of such information.

Under the partnership, which starts this week, Alexa will automatically search the NHS website for relevant information when asked about common illnesses. No money will change hands in the deal, Amazon’s first with the state-backed service,...