South Korea (probably)
Contact tracing is seen as one of the main tools the South Korean government used to control the outbreak in the country.
Contact tracing started early in the country. According to Wikipedia,
In its efforts to fight and contain the virus, South Korea has combined testing with contact tracing.
Infected South Koreans are required to go into isolation in government shelters. Their phones and credit card data are used to trace their prior movements and find their contacts. Those who are determined to have been near the infected individual receive phone alerts with information about their prior movements.
"People that had been near infected people received alerts on their phone" has to be taken in a broad meaning. For example, I leave in Busan, the second city of South Korea (3.5 millions spread on a very large area) and I received daily in February and March alerts about infected that went to areas far from my neighborhood. You can also check on the city's official website (in Korean) the whereabouts of the local infected people: Busan website. You can find an English version of the log of an infected Korean in this BBC article. This website can show you the position of infected people in the past weeks and this website uses geolocalisation to tell you if you had been in close proximity of an infected person. Both are individual initiatives though, but they were widely used in the country.
In addition, a contact tracing app later reinforced this contact tracing strategy. From Business Insider
A government app tracks the location of all new visitors to the country; people who violate quarantine have to wear a location-tracking bracelet;
Visitors here means foreigners and Korean returning to their country from abroad. This proves to be efficient to limit clusters started by imported cases. An example: a South Korean that recently came back to the country visited several night-clubs in the entertainment district of the capital city and obviously came into close proximity with many people. From Business Insider,
After the country relaxed its social distancing as the number of new daily cases dropped into the single digits, several people who visited nightclubs in Seoul tested positive for the virus. [...]
The country's response is a how-to in contact tracing. Seoul's mayor shut down all bars and nightclubs in the city. Within two weeks, South Korean officials had tracked down 46,000 people who had contact with known infected partygoers and tested them for COVID-19.
Of course, we cannot be sure how much impact the app had. This app is just one tool of a strategy of contact tracing, combined with other elements (early numerous and cheap test, transparency and information of the population,...) in a global strategy.