American president is too powerful to deal with since he controls both the civilian and the military side.
The chance of the monarch overriding said request is less than 1%.
Even then, parliament is sovereign. Whilst the logistics are complicated due to how things are introduced to the house, if parliament says no to a prime ministers decision, it overrides anything the prime minister who has no absolute power like a president does.
Personally I love the idea that the codes for nukes are surgically implanted in a volunteer, and in order to issue the order to fire the nukes, the CIC must personally carve the codes from that person's chest with a knife, killing them in the process. Or the variant on that idea, that the codes are implanted in their own forearm, and to order the nukes they must cut the codes from out of their own flesh.
We could do the same for all military deployment orders.
The government, unilaterally, against the country's prevalent feelings towards this illegal war of aggression, permitted USA to use British bases, and if I'm not mistaken, without as much as the parliament vote.