There's actually a larger history to our "American cheese" than you would imagine. Also known as "government cheese", it was part of a federal program dating back to the late 1940s. The "cheese product" was developed as a method of long term storage for dairy product bought by the US government to prop up a failing dairy industry. To avoid the high cost of refrigeration, the USDA converted natural caves into cheese storage (cheese caves) that still exist to this day.
In the early 1980s, the U.S. government distributed some 300 million pounds of pungent‑smelling processed cheese that had been produced with federal funds.
www.history.com
In the 1980s the US government offloaded some 300 million pounds of this low cost cheese product into the US market, which is why every school lunch program, jail, government cafeteria, and fast food restaurant had menus laden with this melty "cheese". (Coincidentally, it may have also been the start of America's obesity epidemic.)
I remember my grandmother used to always keep an innocuous brown box of "US government cheese product" in her fridge and would use it to make grilled cheese sandwiches for us. It was horribly salty and tasted nothing like cheese, but it melted like a charm.
These days I'll go for a nice French brie if I want a good melt, but I'm sure that would offend your delicate British sensibilities as well.