What is the type of cheese in "chips and cheese" UK fast-food? (with photos)
The cheese you saw was, if it was real cheese at all, probably cheddar or just plain white cheese. Whatever they put on was not high-quality stuff - takeaways are not known for lavishing money on ingredients. 

Whether you will be able to get anything similar depends on where you are. American and British cheddars have different flavors, so if I was in the US I'd try Monterey Jack, or maybe Provelone.