Gary Locke served as the governor of the US state of Washington from 1997 to 2005. Locke won 58% of the popular vote in the 1996 Washington gubernatorial election
and was re-elected for a second term in 2000, again with 58% of the votes.
To my knowledge, he was the first person of Chinese descent to hold the office of governor in any US state. He subsequently also served as Secretary of Commerce during the Obama administration from 2009 to 2011. The Secretary of Commerce ranks 10th in the presidential line of succession.
Locke is a third-generation Chinese American with roots in Hong Kong and Guangdong who reportedly did not speak English until he was five years old.
In the 2020 census 5.2 million people out of a total US population of 332 million claimed Chinese ancestry, or about 1.6%. This relative low percentage is in part due to the Chinese Exclusion Act which was in effect from 1882 until 1943, a direct consequence of the strong racism directed towards Chinese people at the time [1]. Major immigration of Chinese people to the United States therefore is a relatively recent phenomenon. As a consequence one would expect people of Chinese descent serving in government mostly in states where they represent a larger percentage of the population.
For example, in California (where Chinese Americans represent 3.6% of the population) there is currently one Chinese-American politician serving as an elected officer of the executive branch, state treasurer Fiona Ma. This office has in the recent past also been held by Chinese Americans John Chiang and Matt Fong. The elected office of state controller has in the recent past been held by Chinese-American politicians Betty Yee and John Chiang. While California is obviously a US state and not a country, it has almost 40 million inhabitants and the size of its economy exceeds that of the United Kingdom.
[1] For background reading:
Iris Chang, The Chinese in America: A Narrative History, Penguin 2004
Jean Pfaelzer, Driven Out: The Forgotten War against Chinese Americans, University of California Press 2008
A. Bandlamudi, "San Jose Had 5 Chinatowns. What Happened To Them?", KQED online, June 17, 2021
D. B. Taylor, "San Jose Apologizes for Decades of Discrimination Against Chinese", The New York Times online, September 30, 2021