The Transatlantic accent, also called a Mid-Atlantic accent, is a way of speaking English that is halfway between American and British. It makes you sound like you have a good education but no one can tell quite where you are from. You hear it in old Hollywood films from the 1930s and 1940s. It is the accent of Cary Grant, Katherine Hepburn, William F Buckley and (at least in some films) God.
There is no town in the world where people grow up speaking English that way. Instead you get the accent in one of three ways