How or why did that dissuade the USA from doing anything?
The scenario was that, under Flexible Response, the Soviets and the Americans might get involved in a fighting war in Europe, up to and involving the use of tactical nukes, but staying below the strategic weapon level by tacit mutual agreement.
However, the UK deterrent used/uses the same missiles as the US deterrent and patrols in the same mid atlantic zones.
What has to be considered under this scenario is that if the UK were hit by Soviet tactical nuclear weapons, they would respond on Russia with a Polaris launch (Moscow Option). Not being able to distinguish this from a US launch and possible decapitation first strike, the Soviets would have to launch - triggering a full scale exchange.
Basically, the UK had the possibility of triggering a full scale exchange, therefore both major powers had an incentive to, at worst, stay non-nuclear and, as it turned out, not fight at all.