They'd be likely to get carved up in a back-room deal with Russia and NATO. The US would immediately declare war on Turkey and begin destroying its cities and infrastructure. The Turkish air force would be completely disabled within a week. All major access points to Turkey's cities, all transit lines, would be bombed and disabled, shutting down their economy and supply lines. All power generating stations and major grid lines would be disabled or bombed and non-functioning within the first week. Air superiority would be accomplished rapidly, the US could sit outside of Turkey and hit major targets with cruise missiles at will.
So what was your premise again, the US would lose some troops? Well that happens in a war. Is the premise that Turkey would threaten to use or attempt to use the nukes on US allies? All that would accomplish is providing justification for either preemptively nuking Turkey to put an end to the war, or dramatically escalating the all-out attack on Turkey to attempt to cripple them faster and convince the military to turn on Erdogan (which would happen very quickly). There's no scenario in which Turkey is a meaningful threat for long.
