US generals actively considered the use of Atomic Bombs from the very beginning, even before China intervened. US presidents considered the use of the A-Bombs after PVA entered.

[From Blair]

On June 1950, Eisenhower met with Collins, Haislip, Ridgway, Ike suggested use of two atomic bombs in the Korea area.

In July 1950, MacArthur suggested plan to use atomic bombs to 'isolate the battle fields".

[From Hastings]

On November 30 1950, President Truman said in a press conference: "There had always been active consideration of its[Atomic Bomb's] use...".

On December 24 1950, MacArthur submitted a list of 'retaliation targets' in China and North Korea, requiring 26 atomic bombs.

In January 1953, US tested its first tactical nuclear weapon, and the JCS considered its use "against military targets affecting operations in Korea."'

In February 1953, in a NSC meeting, President Eisenhower suggested the Kaesong area of North Korea as an appropriate demonstration ground for a tactical nuclear bomb--it "provided a good target for this type of weapon".

On May 19 1953, the Joint Chiefs recommended direct air and naval operations against China, including the use of nuclear weapons. The National Security Council endorsed the JCS recommendation the next day.

Dulles, the Secretary of State was visiting India and told Nehru to deliver a message to Zhou Enlai: if peace was not speedily attained, the United States would begin to bomb north of Yalu, and US had recently tested atomic shells.

33. As a side question, did US threaten China with nukes after the Korean war?

Yes.

US threatened China with nuclear weapons again in 1959. 

From recently declassified documents, President Kennedy considered using nukes to bomb Chinese nuclear facilities in early 1960s , when China was on the verge of exploding its own bomb, but JFK was assassinated and the plan was dropped by President Johnson.

Facing nuclear threat, Chairman Mao said:"we need to have some atomic bombs too". In 1964, China exploded its first A-Bomb. 30 months later, in 1967, it exploded its first H-Bomb. Since then, China has developed a variety of strategic and tactical weapons. China also produced missiles of various ranges, initially targeting US bases at Japan and Philippines, and eventually the North America continent. Mao also said:"We must have nuclear submarines even if this would take us ten thousand years". China tested its nuclear subs in early 1970s and tested SLBMs later. The exact size of PLA nuclear stockpile is unknown, but reasonable estimate put it in the range of 2000-4000 warheads.

In March 1996, PLA conducted an exercise in the Taiwan Straits, President Clinton sent two carriers to the straits. PLA responded by dispatching its nuclear attack submarines and the US fleet stayed 300 nautical miles off Taiwan. In the meantime, PLA SAF (Secondary Artillery Force) conducted exercise to retaliate against enemy strategic strikes. PLA Vice Chief of Staff,  Gen. Xiong Guangkai reportedly hinted that US cares more about LA than Taiwan.