By the 1980s China had land-based ICBMs, such as the DF-5 capable of reaching London. So threatening China with nuclear weapons would not have been productive for the UK unless the US had been willing to partake in the threat with its own massive arsenal.
The UK did consider convincing the US to use nuclear weapons against China some decades prior, and in relation to defending Hong Kong, even against a conventional Chinese attack, which even then was considered indefensible. Alas, the US was rather unreceptive:
In 1961 the UK felt nuclear retaliation was the only alternative to abandoning its colony if China attacked.
Officials wanted "to encourage" China to believe nuclear action against it would follow any hostile action, papers released by the National Archives show.
Britain handed Hong Kong back to China in 1997 having held it since 1842.
The letters, circulated to prime minister Harold Macmillan, were written between 1957 and 1961 when concern was growing about China's intentions.
Hong Kong was thought to be vulnerable, particularly as water and food supplies, from the mainland, could be cut off at any time.
The letters do not say who initially suggested the nuclear option, but they do show that British officials were keen for the Chinese to be aware of the threat while not giving the impression that Hong Kong was an American military outpost.
Defence minister Harold Watkinson wrote to the foreign secretary and prime minister, saying: "Our object is to encourage the Chinese to believe than an attack on Hong Kong would involve US nuclear retaliation."
Foreign Secretary Sir Alec Douglas-Home wrote a "top secret" letter to Mr Watkinson and Mr Macmillan in February 1961.
He wrote: "It must be fully obvious to the Americans that Hong Kong is indefensible by conventional means and that in the event of a Chinese attack, nuclear strikes against China would be the only alternative to complete abandonment of the colony.
"In these circumstances it is perhaps not so much formal staff talks with the Americans that we need so much as an informal exchange of views involving a discussion of the use of nuclear strikes.
"I need hardly say, however, that I agree entirely with your view that while we should encourage the Chinese to believe that an attack on Hong Kong would involve nuclear retaliation, we must avoid anything that would allow the Chinese to claim that the Colony is a military outpost of the United States."
Secret meetings between British and American officials were held in Hawaii with the possibility of further meetings on board a US naval carrier during its frequent visits to Hong Kong.
However, the idea stalled after Admiral Harry Felt, the commander of the US Pacific Fleet, responded cautiously when it was put to him.
Any further response from the US at the time was not included in the National Archives file.
For more see https://www.jstor.org/stable/40111055
In 1958, the garrison
consisted of eight units, two of them on temporary assignment, which
'could not defend the colony against open aggression for the 48 hours or
more needed before any American nuclear weapons could be brought to
bear', given 'the delay inherent in obtaining political approval' for the use
of nuclear weapons from the president. But as the Eisenhower administration
had never committed the United States to the defence of Hong Kong,
the decision for 'massive nuclear retaliation' against Chinese aggression
would depend on the political circumstances. In reality, Hong Kong
remained indefensible [...]
the British, owing to their recognition by 1961 of Hong Kong's
indefensibility, no longer sought US help. The termination of the Hong
Kong Working Group convinced the Kennedy administration that the
United Kingdom did not expect the United States to play a direct role in
Hong Kong's defence. This assessment also governed the Lyndon B.Johnson
administration's response in 1967, when riots threatened Hong Kong's
survival as a crown colony.
Also, by the 1970, during the Nixon administration, the US had started a rapprochement with China, which meant that a maverik UK position on Hong Kong was even less likely to get American support...