The short answer: Yes, Koreans were conscripted starting in 1944, and Formosans (Taiwanese) starting in 1945.
The longer answer:
Koreans were first allowed to volunteer for military service in 1938, and Formosans in 1942. The timing for Koreans was due to the combination of the two things: the Second Sino-Japanese War began in 1937, greatly increasing the Japanese need for military manpower, and Japanese efforts to Japanify Korea kicked into high gear in 1937. The opportunity to serve in the Japanese armed forces was part of the plan to make Koreans into loyal citizens of the Japanese Empire.
By the end of 1941, almost 250,000 applications for military service had been received from Koreans, increasing to over 800,000 by the end of 1943. This is not entirely representative of the desire of Koreans to fight for Japan, since colonial authorities encouraged applications by threats and bribery. These 800,000 applications translated into just under 17,000 enlistments - well over a half-million potential soldiers were turned away, at a time when Japan was short of manpower. There appears to have been some reluctance to train and arm so many potential Korean rebels - the Japanese wanted to rule over a docile and subservient colonial population, not The majority of those who enlisted went into labour units and other rear-area units rather than front-line units.
Japan announced the coming conscription of Koreans in 1942, but delayed implementation because (a) they felt more Japanisation was necessary before Koreans could be trusted, (b) too few Koreans spoke Japanese well, and (c) the family register system was sufficiently outdated and chaotic that the colonial authorities had little idea of who could be conscripted. Thirty months later, they felt that improvement had been made in all three areas, and the need for manpower was growing ever greater, and conscription of Koreans began. The language problem was partly addressed by a pre-military training program for Korean youths which included Japanese language; about 210,000 Koreans passed through this training program.
The first round of Korean conscripts went into the army and navy (45,000 and 10,000 men respectively) in December 1945, and the second (and last) round, also 45,000 and 10,000 in May 1945. More conscription was planned - over 270,000 draft notices had been sent out, and over 230,000 returned. Another 22,000 Koreans were conscripted into labour units.
Over 400,000 Formosans volunteered for military service. As with Korea, there was often coercion involved. 1,000 of the volunteers were accepted. Once conscription began, conscripts greatly outnumbered those accepted as volunteers - overall, about 200,000 Formosans served in military and labour units. Of them, about 30,000 died.
Korean and Formosan labour units often received harsh treatment from the Japanese troops they were working for, and suffered losses from combat, disease, and starvation. For example, most of the Koreans in labour units sent to New Guinea during the war died (as did most of the Japanese soldiers sent there). Koreans and Formosans in the armed forces rarely rose above the lowest ranks. While given the great privilege of fighting and dying for the Emperor, they were not given even close to equal treatment with the Japanese. Mistreatment of labourers notably increased anti-Japanese sentiment, and this, in combination with widespread military training of their barely-trusted colonial subjects could easily have been a seed of future trouble in the colonies. However, Japan's loss of the war and the end of Japanese rule over Korea and Formosa made this a non-issue.
Koreans and Formosans were also conscripted for civilian labour in Japanese industry (in coal mines and factories). About 650,000 Korean workers were sent to Japan, and about 2,500,000 worked for the Japanese within Korea. These civilian workers are not included in the conscription figures above; the labour units noted were working for the military directly, rather than in industry.
Further reading:
Palmer, B. (2007), "Imperial Japan's Preparations to Conscript Koreans as Soldiers, 1942-1945", Korean Studies 31, 63-78. http://www.jstor.org/stable/23720161
Louzon, V. (2018), "From Japanese Soldiers to Chinese Rebels: Colonial Hegemony, War Experience, and Spontaneous Remobilization during the 1947 Taiwanese Rebellion", The Journal of Asian Studies 77(1), 161-179. https://doi.org/10.1017/S0021911817001279