Actually, the US have several layers of veto on Trident, from the hardware, through the software and on upwards. For example, the system simply will not accept an independant attack plan (ie, one adopted by NATO, which means one adopted by the US) and even if such a plan could be forced into the system, the hardware can be locked by the Pentagon at any time. In practise, the system is probably always locked with a special clause to allow test firings, which it is capable of distinguishing. That's all I know about but since the source code for the software is not shared with the UK, there could be absolutely anything else in there too.
That's just the way it is, sorry.
America is not our ally and never was, WE are their ally, following them around like the school bully's sidekick hoping that being sycophantic will spare us their wrath. They would never and have never hesitated to shaft us when it suits them. To say that they are any different from Russia or China is foolhardy and ignorant. All three are superpowers pursuing their goals which are chiefly the control of energy supplies to their economies by a combination of manipulating international treaties, assassination-based regime change, and propaganda-based regime change. Democracy and all the rest of the crap spouted by their politicians is just vote fodder for the people back home.