Our militaries are inextricably linked at this point. But it would be the immediate capture and seizure of all American personnel in the country, then a rapid return of British troops to national soil. The EU and Commonwealth would join us, and we would smash the Americans. Their morale would be zero, half of them would desert/rebel/defect and they'd be fighting a world war on multiple fronts. We would just have to hope Trump doesn't press the red button. In that situation that would trigger a second strike of UK & French nukes straight back at them, and the USA and Europe would be glass. Russia and China achieve world domination.
Regarding Trident, that would be an incredibly sticky situation. Immediately our deterrent would be diminished, long term (2-3 years) it would be gone. There would have to be lightning fast deal made with the French, full refit of the subs and warheads for their missiles.