A solid century. According to conventional wisdom (i.e., Maddison, whose estimates should always be taken with a pinch of salt), British and American per capita GDPs (PPP) were rather similar, with Britain’s being slightly higher, until Britain fell clearly behind during the years immediately after the two World Wars, thanks, presumably, to its socialist and laborist policies of the time. Bob Allen confirms U.S. real wages were slightly higher than British throughout the 19th century (at least, by his method of measurement), however, Allen presumes the British profit share of income was higher. The population of China, the largest economy in the world in the early 1850s, was around 430 million, standard error 50 million (a contemporary source lists 396 million; trust these statistics only as much as you trust contemporary Chinese statistics) at the time, and only fell the following decade and a half thanks to the Taiping Rebellion. The population did not significantly change throughout the rest of the 19th century. According to Robert Allen, real wages in London in the 1900s were some four and a half times higher than those in Beijing. Assuming this reflects a GDP per capita (PPP) difference, and that the per capita situation was roughly the same as in the U.S. (which is fairly reasonable), the U.S. economy surpassed the Chinese sometime (see U.S. population figures) in the early 20th century, probably by 1908. The significance of labor force participation, population pyramids, and the rent share of income is probably sizable here, but is rather difficult to ascertain. The assumption that they all even out in the end is a somewhat dubious, though not completely unreasonable one. However, the general pattern-that the U.S. economy surpassed China’s in the early 20th century, very likely by 1910-remains very sound. In the case of the U.K., for which population figures are readily available, the case is much clearer. The U.S. surpassed the British economy sometime in the early 1850s, certainly by 1855, and almost certainly by 1854, even if it did have a lower GDP per capita (PPP) at the time.
Edit: a study confirms British GDP per capita was higher than American until 1880, when it reached approximate parity, and that the divergence became larger in America’s favor 1899-1906 and 1920-1926.