LONDON—With a vaccine against the new coronavirus many months away and no known effective drug treatments yet, governments around the world are searching for the next best thing: a test to see if people are immune to the virus.

In Britain, the government said Wednesday it is trialing personal blood-testing kits that it hopes to distribute as soon as next week. The test—if it functions—could clear the way for people who have caught and recovered from Covid-19 to return to work or volunteer to help others, potentially easing...