From Minutes of the UK Defence Select Committee...
UK'S TRIDENT SYSTEM NOT TRULY INDEPENDENT
33. Acquiring Trident gave the UK a greater nuclear weapons capability than it could ever have achieved on its own. This enhanced capacity, however, had significant consequences.
34. The fact that, in theory, the British Prime Minister could give the order to fire Trident missiles without getting prior approval from the White House has allowed the UK to maintain the façade of being a global military power. In practice, though, it is difficult to conceive of any situation in which a Prime Minister would fire Trident without prior US approval. The USA would see such an act as cutting across its self-declared prerogative as the world's policeman, and would almost certainly make the UK pay a high price for its presumption. The fact that the UK is completely technically dependent on the USA for the maintenance of the Trident system means that one way the USA could show its displeasure would be to cut off the technical support needed for the UK to continue to send Trident to sea.
35. In practice, the only way that Britain is ever likely to use Trident is to give legitimacy to a US nuclear attack by participating in it. There are precedents for the USA using UK participation in this way for conventional military operations. The principal value of the UK's participation in the recent Iraq war was to help legitimise the US attack. Likewise the principal value of the firing of UK cruise missiles as part of the larger US cruise missile attack on Baghdad was to help legitimise the use of such weapons against urban targets.
36. The most likely scenario in which Trident would actually be used is that Britain would give legitimacy to a US nuclear strike by participating in it.