Can a U.S. citizen marry a British citizen in the U.S. if they intend on living in the U.K.?

Yes, the British citizen comes over on visa waiver. You can get married. If you both plan to then go to the UK straight after, you will then need to apply for a spouse visa for the US citzen at the UK/British Embassy in the US. This will allow the US spouse to come to the UK and also work straightaway.