Originally Posted by
Often1
There is no linkage, A dual UK / US national traveling US-UK would show his UK passport to the air carrier, but would show the US passport to CBP on arrival to the US. In a third country, take a look at which passport has the least restrictive entry requirements and use that. Some will require you to exit on that passport.
I don't think it would work this way ... the air carrier needs to know you will be allowed to enter the USA and for that they need to see the US passport. If you show them the UK passport they will probably deny boarding unless there is a visa or ESTA attached.