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.
No, that didn't change a damned thing, nothing at all.
The basic American Coid War attitude was to be prepared for nuclear war, full stop, period.
Furthermore, you're saying the UK might launch nukes before the USA does so? Never happen, tday's Britons are much too civilized for that,. and everybody knows it.
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Try this:
"Basically, the USA has the possibility of triggering a full scale exchange, it's hard to tell an American missile launch from a UK missile, therefore Britain has an incentive to, at worst, stay non-nuclear and, as it may turn out, not fight at all."