We imported a pup 3 years ago. It is highly recommended to contact a shipping broker to handle all of the export paperwork and US customs paperwork and help you through the process. We went with PBS International https://www.pbs-int.co.uk
They mail you the paperwork you need for picking him up and they mail paperwork to the breeder or person shipping for their end. The broker and plane tickets cost about $1000 if I remember right.
We used United cargo. Pup has to be minimum 10 weeks old. The broker books the flight for pup and gets a direct flight from Heathrow to your closest major airport. The broker also sends the shipper a proper crate with food and water cups, a funnel for airline staff to fill the water and an absorbent pad and all the right information labels. They have to get pup to Heathrow about 4 hours ahead of flight time, I believe.
You pick up your pup at the United cargo building at your airport and find the customs agent to hand over your paperwork. You have to promise to quarantine pup for a certain period to ensure he doesn't have rabies. The quarantine can take place at your home.
Lanie, Polmaise will be along shortly with the "Bobby Moore" (411) if he hasn't already PM'd you - he's sent a few our way over the years. In the olden days (1980s-90s) easiest pup transport was to go across and bring back a pup as carry-on. My last one came over in utero with a veterinian acting as courier. Best thing I'd advise until Polmaise fills you in is get to know the top UK folks in the particular breed you're wanting to import, as they're usually very happy to work with you and get one of their pups into the Yanks' hands. I can say that with absolute certainty for the "minor spaniels" and the working terriers. Or you could go the "cousin" route - my AA "British Lab" bitch - AA being all-age, not All-American - came by way of Nebraska. Same could stated for many field-bred cockers and springers as "seconds generation" British imports.