If your passport is less than 12 months old, you can get one in your new name for free. Otherwise, you will have to pay for a new passport. You can get the new passport from the American consulate in London, it should take around a week.
However, in the UK, you can just start using your married name, you don't need to do anything official. Your marriage certificate will be enough to prove the name change to most people.
The "leave to remain" that you want is FLR(M). You can apply in person if your application is straightforward, or pay £200 less to apply by mail. That could take up to 14 weeks to process, however, so if you must travel in June and you must change your passport, then you'll have to apply in person.
One small thing, though, and I hope the experts will correct me if I'm wrong, but I think you might run into trouble because you reentered the country on a fiance visa that was technically no longer valid--because you're not a fiance anymore, you're a wife. You might have to apply for the spousal visa from the US. I hope someone will come along soon and speak with more authority on the subject.