My comments from a previous thread:

Taiwan did look into developing nuclear weapons. They had a nuclear power program in the 1950s as part of Atoms for Peace, and had a bilateral "peaceful" nuclear agreement with the USA. After the Chinese tested their first nuclear weapon in 1964 they set up a covert exploratory program for looking into nuclear weapons development ("Hsin Chu Program"). In the 1970s they got even more interested in this as the US under Nixon got friendly with the PRC. They bought a heavy water reactor from Canada with the possibility of using it for generating plutonium, and starting filling out other angles of the fuel cycle. The US noticed all of this and basically told them to stop. Taiwan bowed under this pressure and shut down the problematic facilities (a reprocessing facility under development, which would allow them to use their reactor for military purposes) in 1978. But in the 1980s they restarted some of their interest again after the US annulled the Mutual Defense Treaty (as another friendly move with the PRC). This ended in 1988 after the US again got wind of it (this time from a defector), and the Taiwanese shut down their reactor. As far as anyone knows they have not tried it again.

Many sources on this (Richelson's Spying on the Bomb is your best go-to, one-stop shop discussion of small historical nuclear programs), but the above account is largely derived from the very nice summary in Monte Bullard and Jing-dong Yuan, "Taiwan and Nuclear Weaponization: Incentives versus Disincentives," in William C. Potter and Gaukhar Mukhatzhanova, eds., Forecasting Nuclear Proliferation in the 21st Century, Volume 2 (Stanford University Press, 2010), chapter 8.

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To address more directly the "how close" question — not that close. They were assembling the pieces of what could have been a weapons program when it was stopped.